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THE BINARY STARS
BY ROBERT G. AITKEN
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{continued on inside back cover)
{Frontispiece^
Plate I. — The 36inch refractor of the Lick Observatory,
THE BINARY STAES
BY
ROBERT GRANT AITKEN
Late Director and Astronomer, Lick Observatory,
University of California
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright ©1935 by the McGrawHill Book Company,
Inc.
Copyright renewed 1963 by Malcolm D. Aitken
Copyright ©1964 by Dover Publications, Inc.
All rights reserved under Pan American and
International Copyright Conventions
Published simultaneously in Canada by
McClelland and Stewart, Limited
Published in the United Kingdom by
Constable and Company, Limited,
10 Orange Street, London W.C.2
This Dover edition, first published in 1964, is an
unabridged and corrected republication of the second
edition, published by the McGrawHill Book Company
in 1935
The publishers wish to thank Jack T. Kent, Associate
Professor of Mathematics and Astronomy at Texas
A & M University, for preparing the corrections and
supplementary reference materials that have been
incorporated into this Dover edition.
Library of Congress Catalog Card Number: 6413456
Manufactured in the United States of America
Dover Publications, Inc.
180 Varick Street
New York 14, N.Y.
PREFACE TO THE DOVER EDITION
During the past several years it has become more and more
apparent that a need exists for the publication of a new edition
ofAitken's The Binary Stars. Since the book has been out of
print for many years, copies of it are practically unattainable.
With the advent of the Space Age, many of the methods presented
in this book are applicable to other fields than binary stars, and many
of the binarystar astronomers would either like to replace their old
wornout copies, or own one for the first time. It is with the idea
of serving these people, as well as libraries, computing centers, and
industry, that this edition is presented.
No attempt has been made to bring the material up to date. This
has been adequately done elsewhere, as is indicated throughout this
book by new references. The book is simply presented as a classic
in the field. We have attempted to eliminate all known errors, and
to present sufficient additional footnotes and references to start the
reader on the right path should he wish to continue his reading and
study.
We wish to express our appreciation to the following people for
their suggestions and invaluable assistance in listing all known errors,
and eliminating them: Dr. George Van Biesbroeck, Yerkes Observa
tory; Dr. Hamilton M. Jeffers, Lick Observatory; Dr. W. H. van
den Bos, Union Observatory, Johannesburg, South Africa; Dr. K. A.
Strand, U.S. Naval Observatory; and others.
Jack T. Kent
Texas A & M University
November, 1963
PREFACE
The first edition of this book was prepared as a contribution
to the Series of SemiCentennial Publications issued in 1918 by
the University of California and was included in that series,
although published commercially. It has long been out of
print and is now also, in large part, out of date, as a result
of the great amount of work that has been done in the field of
binary star astronomy in the past sixteen years.
In its chapter headings and in the general form of presentation
the present edition follows the plan adopted in the original work,
but it has been necessary to revise all of the chapters and to
rewrite some of them in large part to take account of the work
done in recent years. One measure of this later work is given
by the tables of orbits in the Appendix. In the first edition
87 orbits of visual binaries and 137 orbits of spectroscopic binaries
were listed. In the present edition, the two tables which are
based upon all data available to me before September 1933,
contain 116 and 326 pairs, respectively, though Cepheids and
pseudoCepheids are excluded.
It is a pleasure to express my gratitude to my colleague,
Dr. J. H. Moore, for his kindness in revising the excellent chapter
(V) on The Radial Velocity of a Star, which he prepared for the
first edition; to Profs. H. N. Russell and R. S. Dugan for criticisms
and suggestions relating to the chapter on Eclipsing Binary Stars,
and to Dr. W. H. Van den Bos for placing data relating to the
ThieleInnes method at my disposal, and to acknowledge my
indebtedness to other friends. I desire also to express again
my thanks to all those who gave generous assistance in the
preparation of the first edition.
Robert Grant Aitken.
University op California,
April, 1935.
CONTENTS
Paob
Preface vi
Introduction ix
CHAPTER I
Historical Sketch: The Early Period 1
CHAPTER II
Historical Sketch: The Modern Period 20
CHAPTER III
Observing Methods, Visual Binary Stars 41
CHAPTER IV
The Orbit of a Visual Binary Star 70
CHAPTER V
The Radial Velocity of a Star, by Dr. J. H. Moore 125
CHAPTER VI
The Orbit of a Spectroscopic Binary Star 151
CHAPTER VII
Eclipsing Binary Stars 181
CHAPTER VIII
The Known Orbits of the Binary Stars 203
CHAPTER IX
Some Binary Systems of Special Interest 235
CHAPTER X
Statistical Data Relating to the Visual Binary
Stars in the Northern Sky 257
CHAPTER XI
The Origin of the Binary Stars 273
Appendix 283
Table of Orbits of Visual Binary Stars 284
Table of Orbits of Spectroscopic Binary Stars . 288
Index 303
vii
INTRODUCTION
It is the object of this volume to give a general account of
our present knowledge of the binary stars, including such an
exposition of the best observing methods and of approved
methods of orbit computation as may make it a useful guide
for those who wish to undertake the investigation of these
systems; and to present some conclusions based upon the author's
own researches during the past forty years.
The term binary star was first used by Sir William Herschel,
in 1802, in his paper "On the Construction of the Universe,"
to designate "a real double star — the union of two stars, that
are formed together in one system, by the laws of attraction."
The term double star is of earlier origin; its Greek equivalent
was, in fact, used by Ptolemy to describe the appearance of
v Sagittarii, two fifthmagnitude stars whose angular separation
is about 14/, or a little less than half of the Moon's apparent
diameter. It is still occasionally applied to this and other
pairs of stars visible to the unaided eye, but is generally used
to designate pairs separated by not more than a few seconds
of arc and therefore visible as two stars only with the aid of a
telescope.
Not every double star is a binary system, for, since all of the
stars are apparently mere points of light projected upon the
surface of the celestial sphere, two unrelated stars may appear
to be closely associated simply as the result of the laws of per
spective. Herschel draws the distinction between the two
classes of objects in the following words:
... if a certain star should be situated at any, perhaps immense,
distance behind another, and but little deviating from the line in which
we see the first, we should have the appearance of a double star. But
these stars being totally unconnected would not form a binary system.
If, on the contrary, two stars should really be situated very near each
other, and at the same time so far insulated as not to be materially
affected by neighboring stars, they will then compose a separate system,
and remain united by the bond of their mutual gravitation toward each
other. This should be called a real double star.
X INTRODUCTION
Within the last half century we have become acquainted
with a class of binary systems which are not double stars at
all in the ordinary sense of the term, for the two component
stars are not separately visible in any telescope. These are
the spectroscopic binary stars, so named because their existence
is demonstrated by a slight periodic shifting to and fro of the
lines in their spectra, which, as will be shown, is evidence of a
periodic variation in the radial velocity (the velocity in the
line of sight, toward or away from the observer) of the star.
With the possible exception of factors which may be introduced
by the fact that the distance between the two components of a
spectroscopic binary is, in general, so small (tidal interactions,
for example), there seem to be no dynamical differences between
the spectroscopic and the visual binary systems. The two classes
will, therefore, be regarded in this volume as members of a
single species.
THE BINARY STARS*
CHAPTER I
HISTORICAL SKETCH: THE EARLY PERIOD
The first double star was discovered about the year 1650
by the Italian astronomer, Jean Baptiste Riccioli. This was
f Ursae Majoris (Mizar). It is a remarkable coincidence that
Mizar was also the first double star to be observed photographi
cally, measurable images being secured by G. P. Bond, at the
Harvard College Observatory in 1857; and that its principal
component was the first spectroscopic binary to be discovered,
the announcement being made by E. C. Pickering in 1889.
In 1656, Huygens saw 6 Ononis resolved into the three prin
cipal stars of the group which form the familiar Trapezium,
and, in 1664, Hooke noted that y Arietis consisted of two stars.
At least two additional pairs, one of which proved to be of
more than ordinary interest to astronomers, were discovered
before the close of the seventeenth century. It is worthy
of passing note that these were southern stars, not visible from
European latitudes — a Cruris, discovered by the Jesuit mis
sionary, Father Fontenay, at the Cape of Good Hope, in
1685, and a Centauri, discovered by his confrere, Father Richaud,
while observing a comet at Pondicherry, India, in December,
1689.
These discoveries were all accidental, made in the course of
observations taken for other purposes. This is true also of the
double stars found in the first threequarters of the eighteenth
century. Among these were the discoveries of y Virginis, in
1718, and of Castor, in 1719, by Bradley and Pound, and of
61 Cygni, by Bradley, in 1753.
No suspicion seems to have been entertained by these astron
omers or by their contemporaries that the juxtaposition of the
two star images in such pairs was other than optical, due to
the chance positions of the Earth and the two stars in nearly
* [See G. Van Biesbroeck, in the book review, Ap. Jour. 82, 368, 1935. —
J.T.K.]
1
2 THE BINARY STARS
a straight line. They were therefore regarded as mere curiosi
ties, and no effort was made to increase their number; nor
were observations of the relative positions of the two com
ponents recorded except in descriptive terms. Father Feuille,
for instance, on July 4, 1709, noted that the fainter star in the
double, a Centauri, "is the more western and their distance is
equal to the diameter of this star," and Bradley and Pound
entered in their observing book, on March 30, 1719, that "the
direction of the double star a of Gemini was so nearly parallel
to a line through k and <r of Gemini that, after many trials, we
could scarce determine on which side of <r the line from *c par
allel to the line of their direction tended; if on either, it was
towards /3."
Halley's discovery, in 1718, that some of the brighter stars,
Sirius, Arcturus, Aldebaran, were in motion, having unmis
takably changed their positions in the sky since the time of
Ptolemy, unquestionably stimulated the interest of astron
omers in precise observations of the stars. These researches
and their results, in turn, were probably largely responsible for
the philosophical speculations which began to appear shortly
after the middle of the eighteenth century as to the possi
bility of the existence of systems among the stars. Famous
among the latter are the Cosmologische Brief e* published in
1761 by Lambert, in which it is maintained that the stars are
suns and are accompanied by retinues of planets. Lambert,
however, apparently did not connect his speculations with the
double stars then known. Six years later, in 1767, John Michell,
in a paper read before the Royal Society of London, presented
a strong argument, based upon the theory of probabilities,
that "such double stars, etc., as appear to consist of two or
more stars placed near together, do really consist of stars placed
near together, and under the influence of some general law,
whenever the probability is very great, that there would not
have been any such stars so near together, if all those that are
not less bright than themselves had been scattered at random
through the whole heavens." Michell thus has the credit of
being the first to establish the probability of the existence of
physical systems among the stars; but there were no observational
* Cosmologische Brief e uber die Einrichtung des WeUbaues, Ausgef ertigt
von J. H. Lambert, Augsburg, 1761.
HISTORICAL SKETCH: THE EARLY PERIOD
data to support his deductions and they had no direct influence
upon the progress of astronomy.
The real beginning of double star astronomy dates from the
activities of Christian Mayer and of Sir William Herschel,
in the last quarter of the eighteenth century. If a definite
date is desired we may well follow Lewis in adopting the year
1779, for that year is marked by the appearance of Mayer's
small book entitled De novis in Coelo Sidereo Phaenominis in
miris Stellarum fixarum Comitibus, wherein he speculates upon
the possibility of small suns revolving around larger ones, and
by the beginning of Herschel's systematic search for double
stars.
The difference between Mayer's speculations and earlier ones
is that his rest in some degree at least upon observations. These
were made with an 8ft. Bird mural quadrant at Mannheim,
in 1777 and 1778. At any rate, in his book just referred to,
he publishes a long list of faint companions observed in the
neighborhood of brighter stars.* As one result of his observa
tions he sent to Bode, at Berlin, the first collection or catalogue
of double stars ever published. The list contained earlier dis
coveries as well as his own and is printed in the Astronomisches
Jahrbuch for the year 1784 (issued in 1781) under the caption
"Verzeichnis aller bisher entdeckten Doppelsterne." The fol
lowing tabulation gives the first five entries:
Unterschied
Gerade
Aufst.
Abwei
chung
Stellung
Grdsse
in der
in der
Abstand
des
Klei
Aufst.
Abw.
G. M.
G. M.
Sec.
Sec.
Sec.
nern
Andromeda beyde 9ter
8 38
29 45N
45
24
46
S. W.
Andromeda beyde 9ter
13 13
2018N
15
29
32
S. O.
f Fische 6. und 7ter
15 33
6 25N
22
9
24
N. 0.
bey n Fische beyde 7ter
19 24
5 ON
4
4
s.
y Widder beyde 5ter
25 22
1813N
3
12
12
s. w.
In all, there are 80 entries, many of which, like Castor and
y Virginis, are among the best known double stars. Others
* This list, rearranged according to constellations, was reprinted by
Schjellerup in the journal Copernicus, 3, 57, 1884.
4 THE BfNAHY STARS
are too wide to be found even in Herschel's catalogues and a
few cannot be identified with certainty. Southern pairs, like
a Ceniauri, are of course not included, and, curiously enough,
6 Orionis is not listed. The relative positions given for the
stars in each pair are little better than estimates, for precise
measures were not practicable until the invention of the "revolv
ing micrometer."
In his comments on Mayer's catalogue, Bode points out that
careful observations of such pairs might become of special
N
\
Sir William Ilerachel.
value in the course of time for the discovery of proper motions,
since it would be possible to recognize the fact of motion in
one or the other star as soon as the distance between them had
changed by a very few seconds of arc. Mayer himself seems
to have had proper motions in view in making his observations
and catalogue rather than any idea of orbital motions.
Sir William Herschel "began to look at the planets and the
stars" in May, 1773; on Mar. 1, 1774, "he commenced his
astronomical journal by noting that he had viewed Saturn's
ring with a power of forty, appearing 'like two slender arms'
and also 'the lucid spot in Orion's sword belt.'" The earliest
double star measure recorded in his first catalogue is that of
& Orionis, on Nov. 11, 1776, and he made a few others in the
HISTORICAL SKETCH: THE EARLY PERIOD 5
two years following. It was not until 1779, however, that he
set to work in earnest to search for these objects, for it was
then that he conceived the idea of utilizing them to test a method
of measuring stellar parallax suggested long before by Galileo.
The principle involved is very simple. If two stars are in
the same general direction from us and one is comparatively
near us while the other is extremely distant, the annual revolu
tion of the Earth about the Sun will produce a periodic variation
in the relative positions of the two. As a first approxima
tion, we may regard the more distant star as absolutely fixed
and derive the parallax of the nearer one from the measured
displacements.
It seemed clear to Herschel that the objects best fitted for
such an investigation were close double stars with components
of unequal brightness. He pointed out in his paper On the
Parallaxes of the Fixed Stars, read before the Royal Society
in 1781, that the displacement could be more easily and cer
tainly detected in a close double star than in a pair of stars
more widely separated and also that in the former case the
observations would be free from many errors necessarily affecting
the measures in the latter.
As soon as I was fully satisfied [he continues] that in the investigation
of parallax the method of double stars would have many advantages
above any other, it became necessary to look out for proper stars. This
introduced a new series of observations. I resolved to examine every
star in the heavens with the utmost attention and a very high power,
that I might collect such materials for this research as would enable
me to fix my observations upon those that would best answer my ends.
In this reasoning, Herschel assumes that there is no physical
connection between the components of such close double stars,
— a fact upon which every writer on the history of doublestar
astronomy has commented. This was not an oversight on his
part, for at the close of his first catalogue of double stars he
remarks :
I preferred that expression [i.e., double stars] to any other, such as
Comes, Companion, or Satellite; because, in my opinion, it is much too
soon to form any theories about small stars revolving round large ones,
and I therefore thought it advisable carefully to avoid any expression
that might convey that idea.
6 THE BINARY STARS
Herschel's telescopes were more powerful than any earlier
ones and with them he soon discovered a far larger number
of double stars than he had anticipated. With characteristic
thoroughness he nevertheless decided to carry out his plan of
examining "every star in the heavens," and carefully recorded
full details of all his observations. These included a general
description of each pair and also estimates, or measures with
the "revolving micrometer," or "lamp micrometer," both
invented by himself, of the apparent distance between the two
components and of the direction of the smaller star from the
larger. The direction, or position angle, of the smaller star,
by his definition, was the angle at the larger star between the
line joining the two stars and a line parallel to the celestial
equator. The angle was always made less than 90°, the letters,
nf, sf, sp, and np being added to designate the quadrant. His
first catalogue, presented to the Royal Society on Jan. 10,
1782, contains 269 double stars, "227 of which, to my present
knowledge, have not been noticed by any person." A second
catalogue, containing 434 additional objects, was presented to
the same society in 1784. The stars in these catalogues were
divided into six classes according to angular separation.
In the first [he writes] I have placed all those which require indeed a
very superior telescope, the utmost clearness of air, and every other
favorable circumstance to be seen at all, or well enough to judge of
them. ... In the second class I have put all those that are proper for
estimations by the eye or very delicate measures of the micrometer. . . .
In the third class I have placed all those . . . that are more than five
but less than 15" asunder; . . . The fourth, fifth, and sixth classes
contain double stars that are from 15" to 30", from 30" to 1' and from
1' to 2' or more asunder.
Class I, in the two catalogues, includes 97 pairs, and contains
such systems as r Ophiuchi, 5 Herculis, e Bootis, £ Ursae Majoris,
4 Aquarii, and f Caned. In general, Herschel did not attempt
micrometer measures of the distances of these pairs because
the finest threads available for use in his micrometers sub
tended an angle of more than 1". The following extracts
will show this method of estimating the distance in such cases
and of recording the position angle, and also the care with which
he described the appearance of each object. The dates of
discovery, or of the first observation, here printed above the
descriptions, are set in the margin at the left in the original.
HISTORICAL SKETCH: THE EARLY PERIOD 7
H. 1. September 9, 1779
€ Bootis, Flamst. 36. Ad dextrum femur in perizomate. Double.
Very unequal. L. reddish ; S. blue, or rather a faint lilac. A very beauti
ful object. The vacancy or black division between them, with 227 is
% diameter of S.; with 460, 1% diameter of L.; with 932, near 2 diame
ters of L.; with 1,159, still farther; with 2,010 (extremely distinct),
% diameters of L. These quantities are a mean of two years' observa
tion. Position 31° 34' n preceding.
H. 2. May 2, 1780
£ Ursae Majoris. Fl. 53. In dextro posteriore pede. Double. A
little unequal. Both w [white] and very bright. The interval with 222
is % diameter of L.; with 227, 1 diameter of L; with 278, near 1)4
diameter of L. Position 53° 47' s following.
Careful examination of the later history of the stars of Her
schel's Class I shows that the majority had at discovery an
angular separation of from 2" to 3"5; a half dozen pairs as
wide as 5" are included (one with the manuscript remark, "Too
far asunder for one of the first class"); and a number as close
as or closer than 1". Seven of these stars do not appear in the
great catalogue of Struve, but five of these have been recovered by
later observers, leaving only two that cannot be identified.
In passing judgment upon the accuracy, or the lack of it, in
Herschel's measures of double stars, it is necessary to hold in
mind the conditions under which he had to work. His reflec
tors (all of his own construction) were indeed far more powerful
telescopes than any earlier ones, especially the "twentyfeet
reflector," with mirror of 18%in. aperture, and the great
"fortyfeet telescope," with its 4ft. mirror. But these telescopes
were unprovided with clockwork; in fact, their mountings were
of the altazimuth type. It was therefore necessary to move the
telescope continuously in both coordinates to keep a star in the
field of view and the correcting motions had to be particularly
delicate when highpower eyepieces, such as are necessary in the
observation of close double stars, were employed. Add the
crude forms of micrometers at his disposal, and it will appear
that only an observer of extraordinary skill would be able to
make measures of any value whatever.
No further catalogues of double stars were published by
Herschel until June 8, 1821, about a year before his death,
8 THE BINARY STARS
when he presented to the newly founded Royal Astronomical
Society a final list of 145 new pairs, not arranged in classes,
and, for the most part, without measures.
After completing his second catalogue, in 1784, Herschel
seems to have given relatively little attention to double stars
until about the close of the century and, though he doubtless
tested it fully, there is no mention of his parallax method in
his published writings after the first paper on the subject. A
thorough review of his double star discoveries which he insti
tuted about the year 1797, with careful measures, repeated in
some cases on many nights in different years, revealed a remark
able change in the relative positions of the components in a
number of double stars during the interval of nearly 20 years
since their discovery, but this change was of such a character
that it could not be produced by parallax.
We have seen that, in 1782, Herschel considered the time
not ripe for theorizing as to the possible revolution of small
stars about larger ones. Probably no astronomer of his own
or of any other age was endowed in a higher degree than Herschel
with what has been termed the scientific imagination; certainly
no one ever more boldly speculated upon the deepest problems of
sidereal astronomy; but his speculations were the very opposite
of guesswork; invariably they were the results of critical analyses
of the data given by observation and were tested by further
observations when possible. Michell, in 1783, applied his
earlier argument from the theory of probabilities to the double
stars in HerschePs first catalogue and concluded that practically
all of them were physical systems; but it was not until July,
1802, that Herschel himself gave any intimation of holding
similar views. On that date he presented to the Royal Society
a paper entitled Catalogue of 500 new Nebulae, nebulous Stars,
planetary Nebulae, and Clusters of Stars; with Remarks on the
Construction of the Heavens, in which he enumerates "the parts
that enter into the construction of the heavens" under 12 heads,
the second being, "II. Of Binary sidereal Systems, or double
Stars." In this section he gives the distinction between optical
and binary systems quoted in my Introduction and argues as to
the possibility of systems of the latter type under the law of
gravitation.
On June 9, 1803, followed the great paper in which he gave
the actual demonstration, on the basis of his measures, that
HISTORICAL SKETCH: THE EARLY PERIOD 9
certain double stars are true binary systems. This paper, the
fundamental document in the history of double stars as physical
systems, is entitled, Account of the Changes that have happened,
during the last Twentyfive Years, in the relative Situation of
Doublestars; with an Investigation of the Cause to which they
are owing. After pointing out that the actual existence of
binary systems is not proved by the demonstration that such
systems may exist, Herschel continues, "I shall therefore now
proceed to give an account of a series of observations on double
stars, comprehending a period of about twentyfive years which,
if I am not mistaken, will go to prove, that many of them, are
not merely double in appearance, but must be allowed to be real
binary combinations of two stars, intimately held together by
the bonds of mutual attraction."
Taking Castor as his first example, he shows that the change
in the position of the components is real and not due to any
error of observation. Then, by a masterly analysis of every
possible combination of motions of the Sun and the compo
nents in this, and in five other systems, he proves that orbital
motion is the simplest and most probable explanation in any
one case, and the only reasonable one when all six are considered.
His argument is convincing, his conclusion incontrovertible,
and his paper, a year later, containing a list of 50 additional
double stars, many of which had shown motion of a similar
character, simply emphasizes it.
This practically concluded Sir William Herschel's contribu
tions to double star astronomy, for his list of 145 new pairs,
published in 1821, was based almost entirely upon observations
made before 1802. In fact, little was done in this field by any
one from 1804 until about 1816. Sir John Herschel, in that
year, decided to review and extend his father's work and had
made some progress when Sir James South, who had inde
pendently formed similar plans, suggested that they cooperate.
The suggestion was adopted and the result was a catalogue of
380 stars, based upon observations made in the years 1821 to
1823 with South's 5ft. and 7ft. refractors, of 3^in. and 5in.
aperture, respectively. These telescopes were mounted equa
torially but were not provided with driving clocks. They
were, however, equipped with micrometers in which the par
allel threads were fine spider lines. The value of the catalogue
was greatly increased by the inclusion of all of Sir William
10 THE BINARY STARS
Herschel's measures, many of which had not before been
published.
Both of these astronomers devoted much attention to double
stars in following years, working separately, however, South
with his refractors, Herschel with a 20ft. reflector (18in. mirror)
and later with the 5in. refractor which he had purchased from
South. They not only remeasured practically all of Sir William
Herschel's double stars, some of them on many nights in different
years, but they, and in particular Sir John Herschel, added a
large number of new pairs. Indeed, so numerous were J. Her
schel's discoveries and so faint were many of the stars that he
deemed some apology necessary. He says,
... so long as no presumption a priori can be adduced why the most
minute star in the heavens should not give us that very information
respecting parallax, proper motion, and an infinity of other interesting
points, which we are in search of, and yet may never obtain from its
brighter rivals, the minuteness of an object is no reason for neglecting
its examination. . . . But if small double stars are to be watched, it is
first necessary that they should become known; nor need we fear that
the list will become overwhelming. It will be curtailed at one end, by
the rejection of uninteresting and uninstructive objects, at least as fast
as it is increased on the other by new candidates.
The prediction made in the closing sentence was not imme
diately verified; on the contrary, as late as 1905 Burnham
included in his General Catalogue of All Double Stars within 121° of
the North Pole every pair published as a double star, even those
which had been rejected by their discoverers when they revised
their lists.
The long series of measures and of discoveries of double stars
by Herschel and South were of great value in themselves and
perhaps of even greater value in the stimulus they gave to the
observation of these objects by astronomers generally, and well
merited the gold medals awarded to their authors by the Royal
Astronomical Society. The measures, however, are now assigned
small weight on account of the relatively large errors of observa
tion due to the conditions under which they were of necessity
made; and of the thousands of new pairs very few indeed have
as yet proved of interest as binaries. The great majority are
too wide to give the slightest evidence of orbital motion in the
course of a century.
HISTORICAL SKETCH: THE EARLY PERIOD 11
The true successor to Sir William Herschel, the man who
made the next real advance in double star astronomy, an advance
so great that it may indeed be said to introduce a new period
in its history, was F. G. W. Struve. Wilhelm Struve became
the director of the observatory at Dorpat, Russia, in 1813,
and soon afterward began measuring the differences in right
ascension and in declination between the components of double
1<\ G. W. Struve.
stars with his transit instrument, the only instrument available.
A little later he acquired a small equatorial, inferior to South's,
with which he continued his work, and, in 1822, he published
his Catalogus 795 stellarum duplicium. This volume is interesting
but calls for no special comment because Struve's great work
did not really begin until two years later, in November, 1824,
when he received the celebrated Fraunhofer refractor.
That telescope as an instrument for precise measurements
was far superior to any previously constructed. The tube was
13 ft. long, the objective had an aperture of nine Paris inches,*
the mounting was equatorial and of very convenient form,
and, best of all, was equipped with an excellent driving clock.
♦This is Struve's own statement. Values ranging from 9M to 9.9 in.
(probably English inches) are given by different authorities.
12 THE BINARY STARS
So far as I am aware, this was the first telescope employed
in actual research to be provided with clockwork though Passe
ment, in 1757, had "presented a telescope to the King [of France],
so accurately driven by clockwork that it would follow a star
all night long." A finder of 23^in. aperture and 30in. focus, a
full battery of eyepieces, and accurate and convenient microm
eters completed the equipment, over which Struve was pardon
ably enthusiastic. After careful tests he concluded that "we
may perhaps rank this enormous instrument with the most
celebrated of all reflectors, viz., Herschel's."
Within four days after its arrival Struve had succeeded in
erecting it in a temporary shelter and at once began the first
part of his well planned program of work. His object was
the study of double stars as physical systems and so carefully
had he considered all the requirements for such an investigation
and so thorough, systematic, and skillful was the execution of
his plans that his work has served as a model to all of his suc
cessors. His program had three divisions : the search for double
stars; the accurate determination of their positions in the sky
with the meridian circle as a basis for future investigations
of their proper motions; and the measurement with the microm
eter attached to the great telescope of the relative positions
of the components of each pair to provide the basis for the
study of motions within the system.
The results are embodied in three great volumes, familiarly
known to astronomers as the Catalogus Novus, the Positiones
Mediae, and the Mensurae Micrometricae. The first contains
the list of the double stars found in Struve's survey of the sky
from the North Pole to —15° declination. For the purposes
of this survey he divided the sky into zones from 73^° to 10° wide
in declination and swept across each zone from north to south,
examining with the main telescope all stars that were bright
enough, in his estimation, to be visible in the finder at a distance
of 20° from the full Moon. He considered that these would
include all stars of the eighth magnitude and the brighter ones
of those between magnitudes 8 and 9. Struve states that the
telescope was so easy to manipulate and so excellent in its optical
properties that he was able to examine 400 stars an hour; and he
did, in fact, complete his survey, estimated to embrace the
examination of 120,000 stars, in 129 nights of actual work in
the period from November, 1824, to February, 1827.
HISTORICAL SKETCH: THE EARLY PERIOD 13
Since each star had to be chosen in the finder, then brought
into the field of view of the large telescope, examined, and, if
double, entered in the observing record, with a general descrip
tion, and an approximate position determined by circle read
ings, it is obvious that at the rate of 400 stars an hour, only
a very few seconds could be devoted to the actual examination
of each star. If not seen double, or suspiciously elongated at
the first glance, it must, as a rule, have been passed over. Struve
indeed definitely states that at the first instant of observation
it was generally possible to decide whether a star was single
or double. This is in harmony with my own experience in
similar work, but I have never been content to turn away
from a star apparently single until satisfied that further exami
nation on that occasion was useless. As a matter of fact, later
researches have shown that Struve overlooked many pairs
within his limits of magnitude and angular separation, and
hence easily within the power of his telescope; but even so
the Catalogus Novus, with its short supplement, contains 3,112
entries. In two instances a star is accidentally repeated with a
different number so that 3,110 separate systems are actually
listed. Many of these had been seen by earlier observers and a
few that had entirely escaped Struve's own search were included
on the authority of Bessel or some other observer.
Struve did not stop to make micrometer measures of his
discoveries while engaged in his survey, and the Catalogus
Novus therefore gives simply a rough classification of the pairs
according to their estimated angular separation, with estimates
of magnitude and approximate positions in the sky based on
the equatorial circle readings. He rejected Herschel's classes
V and VI, taking 32" as his superior limit of distance and divid
ing the stars within this limit into four classes: (1) those under
4"; (2) those between 4" and 8"; (3) those between 8" and 16";
and (4) those between 16" and 32". Stars in the first class
were further distinguished as of three grades by the use of the
adjectives vicinae, pervicinae, and vidnissimae. The following
lines (page 14) will illustrate the form of the catalogue, the num
bers in the last column indicating the stars that had been
published in his prior catalogue of 795 pairs.
The Catalogus Novus, published in 1827, furnished the work
ing program on which Struve's two other great volumes were
based, though the Positiones Mediae includes meridian circle
14
THE BINARY STARS
Nume
rus
Nomen
Stellae
A. R.
Decl.
Descriptio
Num.
C. P.
1
OhO'O
+36° 15'
II (8.9) (9)
2
Cephei 316
0.0
+78 45
I (6.7) (6.7), vicinae
3
Andromedae 31
0.4
+45 25
II (7.8) (10) = H.II 83
1
4
0.9
+ 7 29
II (9), Besseli mihi non
inventa
5
34 Piscium
1.1
+ 10 10
III (6) (10), etiam
Besseli
measures made as early as 1822, and the Mensurae Micromet
ricae some micrometer measures made in the years 1824 to
1828. Micrometer work was not actively pushed until 1828
and fourfifths of the 10,448 measures in the Mensurae were
made in the six years 1828 to 1833. The final measures for the
volume were secured in 1835 and it was published in 1837.
The meridian observations were not completed until 1843, and
the Positiones Mediae appeared nine years later, in 1852.
The latter volume does not specially concern us here, for it
is essentially a star catalogue, giving the accurate positions of
the 2 (the symbol always used to designate Struve's double
stars) stars for the epoch 1830.0. The Mensurae Micromet
ricae, on the other hand, merits a more detailed description,
for the measures within it hold in double star astronomy a
position comparable to that of Bradley's meridian measures in
our studies of stellar proper motions. They are fundamental.
The book is monumental in form as well as in contents, its
pages measuring 173^ by 11 in. It is, as Lewis remarks, not to
be taken lightly, and its gravity is not lessened by the fact that
the notes and the Introduction of 180 pages are written in
Latin. Every serious student of double stars, however, should
read this Introduction carefully.
Looking first at the actual measures, we find the stars arranged
in eight classes, Class I of the Catalogus Novus being divided
into three, to correspond to the grades previously defined by
adjectives, and classes III and IV, into two each. The upper
limits of the eight classes, accordingly, are 1, 2, 4, 8, 12, 16,
24, and 32 seconds of arc, respectively. The stars in each class
are further distinguished according to magnitude, being graded as
lucidae if both components of the pair are brighter than 8.5 mag
nitude, and reliquae if either component is fainter than this.
HISTORICAL SKETCH: THE EARLY PERIOD
15
Sir John Herschel had early proposed that the actual date
of every double star measure be published and that it be given
in years and the decimal of a year. About the year 1828 he
further suggested that position angles be referred to the north
pole instead of to the equator as origin and be counted through
360°. This avoids the liability to mistakes inherent in Sir
William HerschePs method. Both suggestions were adopted
by Struve and have been followed by all later observers. Gen
erally the date is recorded to three decimals, thus defining the
day, but Struve gives only two. The position angle increases
from North (0°) through East, or following (90°), South (180°),
and West or preceding (270°).
The heading of the first section, and the first entry under it
will illustrate the arrangement of the measures in the Men
surae Micrometricae:
Duplices Lucidae Ordinis Primi
Quarum distantiae inter O''OO et l''OO
Epocha
Amplif.
Distant.
Angulus
Magnitudines
2
Cephei 316.
a. = O^'O.
5 = 78° 45'
Major— 6.3 j
} .ava;
mirii
or = 6.6 eerie
flavior
1828.22
600
0''72
342 ?5
6.5,7
1828.27
600
0.84
343.4
6.5, 7
1832.20
600
0.94
339.3
6, 6
1832.24
480
0.70
337.5
6, 6.5
1833.34
800
0.85
344.8
6.5, 6.5m
Medium 1830.85
0.810
341.50
The Introduction contains descriptions of the plan of work,
the instrument, and the methods of observing, and thorough
discussions of the observations. The systems of magnitudes
and of color notation, the division of the stars into classes by
distance and magnitude, the proper and orbital motions detected,
are among the topics treated. One who does not care to read
the Latin original will find an excellent short summary in English
in Lewis's volume on the Struve Double Stars published in
1906 as Vol. LVI of the Memoirs of the Royal Astronomical
Society of London. Three or four of Struve's general conclusions
are still of interest and importance. He concludes, for example,
16 THE BINARY STARS
that the probable errors of his measures of distance are some
what greater than those of his measures of position angle and
that both increase with the angular separation of the components,
with their faintness, and with the difference in their magnitudes.
Modern observers note the same facts in the probable errors of
their measures. In their precision, moreover, and in freedom
from systematic errors, Struve's measures compare very favorably
with the best modern ones.
His observations of star colors show that when the two com
ponents of a pair are of about the same magnitude they are
generally of the same color, and that the probability of color
contrast increases with increasing difference in the brightness
of the components, the fainter star being the bluer. Very few
exceptions to these results have been noted by later observers.
Finally, in connection with his discussion of the division of
double stars into classes by distance, Struve argues, on the
theory of probabilities, that practically all the pairs in his first
three classes (distance under 4" 00) and the great majority in
his first five classes (distance less than 12") are true binary
systems. With increasing angular separation he finds that the
probability that optical systems will be included increases,
especially among the pairs in which both components are as
faint as, or fainter than, 8.5 magnitude. This again is in har
mony with the results from more recent investigations.
The Russian government, in 1839, called upon Struve to build
and direct the new Imperial Observatory at Pulkowa. Here
the principal instruments were an excellent Repsold meridian
circle and an equatorial telescope with an object glass of 15in.
aperture. This was then the largest refractor in the world,
as the 9in. Dorpat telescope had been in 1824.
One of the first pieces of work undertaken with it was a
resurvey of the northern half of the sky to include all stars as
bright as the seventh magnitude. In all, about 17,000 stars
were examined, and the work was completed in 109 nights of
actual observing between the dates Aug. 26, 1841, and Dec. 7,
1842. The immediate object was the formation of a list of
all the brighter stars, with approximate positions, to serve
as a working program for precise observations with the meridian
circle. It was thought, however, that the more powerful
telescope might reveal double stars which had escaped detection
with the 9in. either because of their small angular separation
HISTORICAL SKETCH: THE EARLY PERIOD 17
or because of the faintness of one component. This expecta
tion was fully realized. The survey, which after the first
month was conducted by Wilhelm Struve's son, Otto, resulted
in the discovery of 514 new pairs, a large percentage of which
were close pairs. These, with Otto Struve's later discoveries
which raised the total to 547, are known as the OS or Pulkowa
double stars. The list of 514 was published in 1843 without
measures, and when, in 1850, a corrected catalogue, with meas
ures, was issued, 106 of the original 514 were omitted because
not really double, or wider than the adopted distance limits,
or for other reasons. But, as Hussey says, "it is difficult effec
tively to remove a star which has once appeared in the lists."
Nearly all of the 02 stars rejected because of wide separation
have been measured by later observers and are retained in
Hussey's Catalogue of the OS Stars and in Burnham's General
Catalogue.
The early period of double star discovery ended with the
appearance of the Pulkowa Catalogue. New double stars were
indeed found by various observers as incidents in their regular
observing which was mainly devoted to the double stars in the
great catalogues which have been described and especially
to those in the S and the OS lists. The general feeling, how
ever, was that the Herschels and the Struves had practically
completed the work of discovery.
Many astronomers, in the half century from 1820 to 1870,
devoted great energy to the accurate measurement of double
stars; and the problem of deriving the elements of the orbit
of a system from the data of observation also received much
attention. This problem was solved as early as 1827, and new
methods of solution have been proposed at intervals from that
date to the present time. Some of these will be considered in
Chap. IV.
One of the most notable of the earlier of these observers was
the Rev. W. R. Dawes, who took up this work as early as 1830,
using a 3.8in. refractor. Later, from 1839 to 1844, he had
the use of a 7in. refractor at Mr. Bishop's observatory, and
still later, at his own observatory, he installed first a 6in. Merz,
than a 73^in. Alvan Clark, and finally an 8>^in. Clark refractor.
Mr. Dawes possessed remarkable keenness of vision, a quality
which earned for him the sobriquet, the eagleeyed, and, as
Sir George Airy says, was also "distinguished . . . by a habitual,
18 THE BINARY STARS
and (I may say) contemplative precision in the use of his instru
ments." His observations, which are to be found in the volumes
of the Monthly Notices and the Memoirs of the Royal Astronomical
Society, "have commanded a degree of respect which has not
often been obtained by the productions of larger instruments."
Another English observer whose work had great influence
upon the progress of double star astronomy was Admiral W. H.
Smythe, who also began his observing in 1830. His observa
tions were not in the same class with those of Dawes, but his
Bedford Catalogue and his Cycle of Celestial Objects became
justly popular for their descriptions of the double and multiple
stars, nebulae, and clusters of which they treat, and are still
"anything but dull reading."
Far more important and comprehensive than that of any
other astronomer of the earlier period after W. Struve was the
double star work of Baron Ercole Dembowski who made his
first measures at his private observatory near Naples in 1851.
His telescope had an excellent object glass, but its aperture
was only 5 in. and the mounting had neither driving clock nor
position circles. His micrometer, although it could be rotated,
was not provided with a circle from which the position angle
could be read off. His procedure was to use two parallel fixed
wires separated by a known distance. As the pair was brought
to the first wire, he would set for position angle and measure
with the micrometer thread the position of the primary star on
that wire. Then, letting the star pass to the second wire by the
diurnal motion, he would measure its position on that wire also.
The difference of the two readings and the known distance
between the two wires gave him the two sides of a right triangle
from which the position angle could be computed. With this
instrument Dembowski made some 2,000 sets of measures of
high quality in the course of eight years, though how he managed
to accomplish it is wellnigh a mystery to observers accustomed to
the refinements of modern micrometers and telescope mountings.
In 1859, he secured a 7in. Merz refractor with circles, microm
eter, and a good driving clock, and, in 1862, he resumed his
double star observing with fresh enthusiasm. His general
plan was to remeasure all of the double stars in the Dorpat
and Pulkowa catalogues, repeating the measures in successive
years for those stars in which changes were brought to light.
His skill and industry enabled him, by the close of the year
HISTORICAL SKETCH. THE EARLY PERIOD 19
1878, to accumulate nearly 21,000 sets of measures, including
measures of all of the 2 stars, except 64 which for one reason or
another were too difficult for his telescope. About 3,000 of
the measures pertain to the OS stars and about 1,700 to stars
discovered by Burnham and other observers. Each star was
measured on several different nights and for the more interesting
stars long series of measures extending over 12 or 15 or even
more years were secured. The comprehensive character of his
program, the systematic way in which he carried it into execution,
and the remarkable accuracy of his measures combine to make
Dembowski's work one of the great contributions to double
star astronomy. He died before his measures could be published
in collected form, but they were later (18831884) edited and
published by Otto Struve and Schiaparelli in two splendid
quarto volumes which are as indispensable to the student of
double stars as the Mensurae Micrometricae itself.
Madler at Dorpat, Secchi at Rome, Bessel at Konigsberg,
Knott at Cuckfield, Engelmann at Leipzig, Wilson and Gled
hill at Bermerside, and many other able astronomers published
important series of double star measures in the period under
consideration. It is impossible to name them all here. Lewis,
in his volume on the Struve Stars, and Burnham, in his General
Catalogue of Double Stars, give full lists of the observers, the
latter with complete references to the published measures.
CHAPTER II
HISTORICAL SKETCH : THE MODERN PERIOD
The feeling that the Herschels, South, and the Struves had
practically exhausted the field of double star discovery, at
least for astronomers in the northern hemisphere, continued
for thirty years after the appearance of the Pulkowa Cata
logue in 1843. Nor were any new lines of investigation in
double star astronomy developed during this period. Then,
in 1873, a modest paper appeared in the Monthly Notices of
the Royal Astronomical Society, entitled "Catalogue of Eighty
one Double Stars, Discovered with a Sixinch Alvan Clark
Refractor. By S. W. Burnham, Chicago, U. S. A."
The date of the appearance of this paper may be taken as
the beginning of the modern period of double star astronomy,
for to Burnham belongs the great credit of being the first to
demonstrate and utilize the full power of modern refracting
telescopes in visual observations; and the 40 years of his active
career as an observer cover nearly all of the modern develop
ments in binary star astronomy, including the discovery and
observation of spectroscopic binaries, the demonstration that
the eclipsing variable stars are binary systems, and the applica
tion of photographic methods to the measurement of visual
double stars.
Within a year after the appearance of his first catalogue
Burnham had published two additional ones, raising the num
ber of his discoveries to 182. At that time he was not a profes
sional astronomer but an expert stenographer employed as
official reporter in the United States Courts at Chicago. He
had secured, in 1861, a 3in. telescope with altazimuth mounting,
and, some years later, a 3%in. refractor with equatorial mount
ing. "This was just good enough," he tells us, "to be of some
use, and poor enough ... to make something better more
desirable than ever." In 1870, accordingly, he purchased the
6in. refractor from Alvan Clark and erected it in a small observa
tory at his home in Chicago. With this instrumental equipment
20
HISTORICAL SKETCH: THE MODERN PERIOD
21
and an astronomical library consisting principally of a copy of
the first edition of Webb's Celestial Objects for Common Telescopes,
Burnham began his career as a student of double stars. His
first new pair (£ 40) was found on April 27, 1870.
The 64n. telescope, which his work so soon made famous,
was not at first provided with a micrometer and his earliest
list of discoveries was printed without measures. Later, posi
tion angles were measured, but the distances continued to be
S. W. Burnham.
estimated. This lack of measures by him was covered to a
considerable extent by the measures of Dembowski and Asaph
Hall.
Burnham's later career was unique. He held positions in
four observatories, the Dearborn, the Washburn, the Lick,
and the Yerkes, and discovered double stars also with the
26in. refractor at the United States Naval Observatory,
the 16in. refractor of the Warner Observatory, and the
9.4in. refractor at the Dartmouth College Observatory. In
all, he discovered about 1,340 new double stars and made many
thousands of measures which are of inestimable value because
of their great accuracy and because of the care with which he
prepared his observing programs. And yet, except for the two
22 THE BINARY STARS
short periods spent respectively at Madison and at Mount
Hamilton, he continued his work as Clerk of the United States
District Court of Chicago until 1910! He retired from the
Yerkes Observatory in 1912 and died in 1921.
Burnham's plan in searching for new double stars was very
different from that followed by his great predecessors. He
did not attempt a systematic survey of the sky but examined
the stars in a more random way. In his earlier work, while
identifying the objects described in Webb's book, he made a
practice of examining the other stars near them. Later, when
ever he measured a double star, he continued this practice,
examining in this manner probably the great majority of the
naked eye and brighter telescopic stars visible from our latitudes.
Many of the double stars he discovered with the 6in. refractor
are difficult objects to measure with an aperture of 36 inches.
They include objects of two classes almost unrepresented in the
earlier catalogues: pairs in which the components are separated
by distances as small as 0"2, and pairs in which one component
is extremely faint, and close to a bright primary. In his first
two lists he set his limit at 10", but later generally rejected pairs
wider than 5". The result is that the percentage of very close
pairs, and therefore of pairs in comparatively rapid orbital motion,
is far higher in his catalogue than in any of the earlier ones.
Burnham's work introduced the modern era of double star
discovery, the end of which has not yet come. No less dis
tinguished an authority than the late Rev. T. W. Webb, in
congratulating Burnham upon his work in 1873, warned him
that he could not continue it for any great length of time for
want of material. Writing in 1900, Burnham's comment was:
"Since that time more than one thousand new double stars
have been added to my own catalogues, and the prospect of
future discoveries is as promising and encouraging as when
the first star was found with the sixinch telescope."
Working with the 183^in. refractor of the Dearborn Observa
tory, G. W. Hough discovered 648 double stars in the quarter
century from 1881 to 1906. In 1896 and 1897, T. J. J. See,
assisted by W. A. Cogshall and S. L. Boothroyd, examined the
stars in the zone from —20° to —45° declination, and in half
of the zone (from 4 h to 16 h R. A.) from 45° to 65° declination
with the 24in. refractor of the Lowell Observatory, and dis
covered 500 new double stars. See states that not less than
HISTORICAL SKETCH: THE MODERN PERIOD 23
10,000 stars were examined, "many of them, doubtless, on
several occasions." This is probably an overestimate for it
leads to a remarkably small percentage of discoveries.
In England, in 1901, the Rev. T. E. H. Espin began pub
lishing lists of new double stars discovered with his 17}iin.
reflector. 1 The first list contained pairs casually discovered in
the course of other work; later, Mr. Espin undertook the sys
tematic observation of all the stars in the Bonn Durchmusterung
north of +30°, recording, and, as far as possible measuring,
all pairs under 10" not already known as double. Since 1917,
W. Milburn has been his assistant in this work, which is not
yet completed. In 1932, Espin's published discoveries numbered
2,444 and Milburn's, 673.
Shorter lists of discoveries have been published by E. S.
Holden, F. Kustner, H. A. Howe, O. Stone, Alvan and A. G.
Clark, E. E. Barnard, and others, and many doubles were first
noted by the various observers participating in the preparation
of the great Astronomische Gesellschaft Catalogue.
In France, in 1909, Robert Jonckheere began double star work
at the Observatory of the University of Lille at Hem and in the
course of five years discovered 1,319 new pairs. Forced by
the war to give up his work in France, he went to the Royal
Observatory at Greenwich, England, and for some years con
tinued his observations with the 28in. refractor. The majority
of his double stars, though close, are quite faint, a large per
centage of them falling outside of the 9.5 magnitude limit of
the Bonn Durchmusterung. In 1917 he published a catalogue
of all double stars under 5" discovered visually in the years
1905 to 1916 in the sky area within 105° of the North Pole.
Many pairs, generally wider than 5" and often quite faint,
have been found in the various sections of the Astrographic
Catalogue and listed by Scheiner, Stein, Barton, and others.
My own work in this field of astronomy began when I came
to the Lick Observatory in June, 1895. At first my time was
devoted to the measurement with the 12in. refractor of list
of stars selected by Prof. Barnard, and the work was done under
his direction. Later, longer lists were measured both with
this telescope and with the 36in. refractor; and in selecting the
stars for measurement I had the benefit of advice — so generously
given by him to many double star observers of my generation —
*A 24in. reflector was added later. Mr. Espin died on Dec. 1, 1934.
24 THE BINARY STARS
from Prof. Burnham, then at the Yerkes Observatory. My
attention was early drawn to questions relating to double star
statistics, and before long the conviction was reached that a
prerequisite to any satisfactory statistical study of double star
distribution was a resurvey of the sky with a large modern
telescope that should be carried to a definite limiting magnitude.
I decided to undertake such a survey, and, adopting the magni
tude 9.0 of the Bonn Durchmusterung as a limit, began the
preparation of charts of convenient size and scale showing every
star in the BD as bright as 9.0 magnitude, with notes to mark
those already known to be double. The actual work of compar
ing these charts with the sky was begun early in April, 1899.
Professor W. J. Hussey, who came to the Lick Observatory
in January, 1896, also soon took up the observation of double
stars. His first list consisted of miscellaneous stars, but, in
1898, he began the remeasurement of all of the double stars
discovered by Otto Struve, including the "rejected " pairs. This
work was carried out with such energy and skill that in 1901,
in Volume V of the Lick Observatory Publications, a catalogue
of the OS stars was published which contained not only Hussey's
measures of every pair but also a complete collection of all
other published measures of these stars, with references to the
original publications, and discussions of the motion shown by
the various systems. In the course of this work, Hussey had
found an occasional new double star and had decided that
at its conclusion he would make more thorough search for new
pairs. In July, 1899, we accordingly combined forces for
the survey of the entire sky from the North Pole to —22° declina
tion on the plan which I had already begun to put into execution;
Hussey, however, charted also the 9.1 BD stars. Each observer
undertook to examine about half the sky area, in zones 4° wide
in declination. When Hussey left the Lick Observatory in
1905 to become director of the Observatory of the University
of Michigan, I took over his zones in addition to those assigned
to me in our division of the work and early in 1915 completed
the entire survey to —22° declination, as originally planned,
between 13 h and l h in right ascension, but only to — 14° declina
tion in the remaining 12 h . These come to the meridian in our
winter months when conditions are rarely satisfactory for work
at low altitudes. Subsequently, by agreement with the observers
at the Union Observatory, South Africa, whose work will be
HISTORICAL SKETCH: THE MODERN PERIOD
25
described on a later page, I extended the survey in these 12
hours to — 18° declination.
The survey has resulted in the discovery of more than 4,400
new pairs, 1,329 by Hussey, the others by me, practically all
of which fall within the distance limit of 5". Some statistical
conclusions based upon this material will be presented in a later
chapter.
It may seem that undue emphasis has been placed upon the
discovery of double stars in this historical sketch. That a par
ticular star is or is not double is indeed of relatively little con
sequence; the important thing is to secure accurate measures
through a period of time sufficiently long to provide the data
for a definite determination of the orbit of the system. But
the discovery must precede the measures, as Sir John Herschel
said long ago; moreover, such surveys as that of Struve, the
one recently completed at the Lick Observatory and those in
progress in the southern hemisphere, afford the only basis for
statistical investigations relating to the number and spatial
distribution of the double stars. Further, the comparison of
the distance limits adopted by the successive discoverers of
double stars and an analysis of the actual distances of the pairs
in their catalogues afford the most convenient measure of the
progress made in the 140 years since Herschel began his work,
both in the power of the telescopes available and in the knowledge
of the requirements for advance in this field of astronomy.
The data in the first four lines of the following table are taken
from Burnham's General Catalogue of his own discoveries, and
in the last two lines I have added the corresponding figures for
the Lick Observatory double star survey.
Percentage op Close Pairs in Certain Catalogues ob Double Stars
Class I,
Class II,
Per
number
number
Sum
centage
of stars
of stars
of close
pairs
William Herschel, catalogue of 812 stars .
12
24
36
4.5
Wilhelm Struve, catalogue of 2,640 stars
91
314
405
15.0
Otto Struve, catalogue of 547 stars
154
63
217
40.0
Burnham, catalogue of 1,260 stars
385
305
690
55.0
Hussey, catalogue of 1,327 stars
674
310
984
74.2
Aitken, catalogue of 3,105 stars
1,595
710
2,305
74.3
26 THE BINARY STARS
The increasing percentage of close pairs is, of course, due in
part to the earlier discovery of the wider pairs, but the absolute
numbers of the closer pairs testify to the increase of telescopic
power in the period since 1780. If Class I had been divided into
two subclasses including pairs under 0"50 and pairs between
0"51 and l'.'OO, respectively, the figures would have been even
more eloquent, for 60 per cent of the Class I pairs in the
last two catalogues enumerated have measured distances of / '50
or less.
While the modern period is thus characterized by the number
of visual binaries, and, in particular, those of very small angular
distance discovered within it, it is still more notable for the
development of an entirely new field in binary star astronomy.
In August, 1889, Prof. E. C. Pickering announced that certain
lines in the objectiveprism spectrograms of £ Ursae Majoris
(Mizar) were double on some plates, single on others, the cycle
being completed in about 104 days.* An explanation of the
phenomenon was found in the hypothesis that the star consisted
of two components, approximately equal in brightness, in rapid
revolution about their center of mass.
If the orbit plane of such a system is inclined at a considerable
angle to the plane of projection, the velocities in the line of sight
of the two components will vary periodically, as is evident from
Fig. 1; and, on the DopplerFizeau principle, f there will be a
slight displacement of the lines of the spectrum of each component
from their mean positions toward the violet end when that com
ponent is approaching the Earth, relatively to the motion of the
center of mass of the system, and toward the red end when it is
receding, relatively. It is clear from the figure that when one
component is approaching the Earth, relatively, the other will
be receding, and that the lines of the two spectra at such times
will be displaced in opposite directions, thus appearing double
on the spectrograms. Twice, also, in each revolution the orbital
motion of the two components will evidently be directly across
the line of sight and the radial velocity of each at these times is
the same, and is equal to that of the system as a whole. The
lines of the two spectra, if similar, will then coincide and appear
single on the plates. There is no question but that this explana
* The true period, deduced from many plates taken with slit spectro
graphs, is about onefifth of this value, a little more than 20.5 days,
t Explained in Chap. V.
HISTORICAL SKETCH: THE MODERN PERIOD 27
tion is the correct one, and Mizar therefore has the honor of being
the first star discovered to be a spectroscopic binary system.
A moment's consideration is enough to show that if one of
the two components in such a system is relatively faint or
"dark," only one set of spectral lines, that produced by the
brighter star, will appear upon the plate, but that these lines will
be shifted periodically from their mean positions just as are
the lines in the double spectrum of Mizar. If the plane of the
system lies so nearly in the line of sight that each star partly
or completely eclipses the other once in every revolution, the
To Hie earth
B
Fig. 1. — A, A', A", A'" = primary star at points of maximum, minimum, and
mean radial velocity. B, B', B", B'" = position of the companion star at the
corresponding instants. C is the center of gravity of the system. There is no
star or other body at this point.
presence of the darker star may be revealed by a periodic dim
ming of the light of the brighter one; if the orbit plane, as will
more commonly happen, is inclined at such an angle to the line
of sight that there is no occultation or eclipse of the stars for
observers on the Earth, the variable radial velocity of the brighter
star will be the sole evidence of the existence of its companion.
Algol (j8 Persei) is a. variable star whose light remains nearly
constant about fourfifths of the time; but once in every two
and onehalf days it rapidly loses brightness and then in a few
hours' time returns to its normal brilliancy. As early as 1782,
Goodericke, the discoverer of the phenomenon, advanced the
the ry that the periodic loss of light resulted from the partial
ecli se of the bright star by a (relatively) dark companion. In
No ./ ber, 1889, Prof. H. C. Vogel, who had been photographing
the SJ. 3ctn m of the star at Potsdam, announced that this theory
was correct, for his spectrograms showed that before light mini
28 THE BINARY STARS
mum the spectral lines were shifted toward the red from their
mean position by an amount corresponding to a velocity of reces
sion from the Earth of about 27 miles a second. While the star
was recovering its brightness, on the other hand, the shift of the
lines toward the violet indicated a somewhat greater velocity of
approach, and the period of revolution determined by means of
the curve plotted from the observed radial velocities was identical
with the period of light variation. Algol thus became the second
known spectroscopic binary star and the first of the special class
later called eclipsing binaries.
Within a few months two other spectroscopic binary stars
were discovered; j8 Aurigae by Miss Maury at the Harvard
College Observatory from the doubling of the lines in its spec
trum at intervals of slightly less than two days (the complete
revolution period is 3.96 days), and a Virginis, by Vogel. The
latter star was not variable in its light, like Algol, nor did its
spectrum show a periodic doubling of the lines,* like Mizar and
8 Aurigae, but the lines of the single spectrum were displaced
periodically, proving that the star's radial velocity varied, and
the cycle of variation was repeated every four days, a Virginis
is thus the first representative of that class of spectroscopic
binary systems in which one component is relatively dark, as
in the case of Algol, but in which the orbit plane does not
coincide with the line of sight. It is to this class that the great
majority of spectroscopic binary stars now known belong.
The reader must not infer that the companion stars in systems
of this class emit no light; the expression relatively dark may
simply mean that the companion is two or three magnitudes
fainter than its primary. If the latter were not present, the
companion in many systems would be recognized as a bright
star; even the companion of Algol radiates enough light to permit
the secondary eclipse, when the primary star is the occulting
body, to be detected by our delicate modern photometers.
The story of the modern spectrograph and its revelations of
the chemical composition of the stars and nebulae and of the
physical conditions that prevail in them is a marvelous one,
but this is not the place to tell it. We must limit ourselves to
the simple statement that in the years since 1889 the spectro
graph has also given us a vast amount of information with
* The secondary spectrum of a Virginia has been photographed in more
recent years.
HISTORICAL SKETCH: THE MODERN PERIOD 29
regard to the radial velocities of the stars and, as a byproduct,
with regard to spectroscopic binary systems. In this develop
ment the Lick Observatory has taken a leading part, for by the
application of sound engineering principles in the design of the
Mills spectrograph, and by patient and skilful experimental
work extended over several years, Dr. Campbell was enabled,
in the late 1890's, to secure an accuracy of measurement of
radial velocity far surpassing any previously attained. The
New Mills spectrograph, mounted in 1903, led to even better
results, and it is now possible, in the more favorable cases, to
detect a variation in the radial velocity even if the range is only
1.5 km/sec* Other observers and institutions have also been
most active and successful, and the number of known spec
troscopic binaries has increased with great rapidity. The First
Catalogue of Spectroscopic Binaries, compiled by Campbell and
Curtis to include the systems observed to Jan. 1, 1905, had
140 entries; by Jan. 1, 1910, when Campbell prepared his Second
Catalogue of Spectroscopic Binary Stars, the number had grown
to 306; the Third Catalogue, compiled by Dr. J. H. Moore in
1924 had 1,054 entries, and in December, 1931, the card catalogue
which is kept up to date at the Lick Observatory listed 1,340 stars
with known variable radial velocity and 122 more in which
variation was indicated or suspected. In the vast majority of
cases the variation in radial velocity was detected at obser
vatories in the United States and Canada or at the Chile Station
of the Lick Observatory.
The institutions that have engaged most actively in the meas
urement of stellar radial velocities and the consequent discovery
of spectroscopic binary stars, are the Lick (with its branch station
at Santiago, Chile, until 1929), the Yerkes, the Mount Wilson,
the Dominion Astrophysical (Victoria), and the Dominion
(Ottawa) Observatories. Other observatories in the United
States, those at Pulkowa, Potsdam, and Bonn, in Europe, and
the Cape Observatory, in South Africa, have also made notable
contributions in this field.
The discoveries of the spectroscopic binary stars are ordinarily
credited to observatories rather than individuals because it is
often a matter for fine discrimination to decide with whom the
* This is six times the probable error of measurement of the best plates.
No one, however, would announce a radial velocity of so small a range on
the basis of only two or three plates.
30 THE BINARY STARS
credit for a particular discovery should rest. In general, at
least three spectrograms are required to prove that a star is a
spectroscopic binary star. These may all be taken and meas
ured by a single observer, or the three plates may be exposed
by as many different observers in the course of carrying out
a program of work planned by a fourth; the plates may be
measured by one or more of the four or by others; variation in
the radial velocity may be suspected from the second plate and
confirmed by the third or only by a fourth or still later plate.
Not all stars showing variable radial velocity are spectroscopic
binary stars. When, as in the case of Mizar, or of j8 Aurigae,
two sets of lines appear upon the spectrogram, it is certain that
we have to do with a double star system. When only one set
of lines appears, but this set exhibits a periodic variation, as in
the case of a Virginis, it is almost certain that the light producing
the spectrum comes from the brighter component of such a
system. But in some cases — the Cepheid variable stars, for
example — the apparent variation in radial velocity may be the
consequence of rhythmic or pulsating motion in the atmosphere
of a single star, rather than of the orbital motion of a component
in a binary system. Such stars, and also those in which the
observed variation in radial velocity is quite irregular, will be
considered later.
The problem of finding the elements of the orbit of a spec
troscopic binary from the data given by the measures of radial
velocity was solved as early as 1891 by Rambaut, and in 1894,
LehmannFilh6s published the method which has been the
chief one used ever since. A number of other methods have
been proposed in more recent years, some analytical, others
graphical, and doubtless others still will be developed. This
phase of the subject is treated in Chap. VI.
At the present time orbits for more than 320 systems have been
computed, a number greatly exceeding that of the visual binary
systems with known orbits. The reason is not far to seek.
The visual binaries are systems of vast dimensions and their
revolution periods range from a minimum (so far as known at
present) of 4.6 years to a maximum that is certainly greater
than 1,000 years. Castor, for example, was one of the first double
stars to be observed, and it was the one in which the fact of
orbital motion was first demonstrated; but although the observa
tions extend from the year 1719 to date, the length of the revolu
HISTORICAL SKETCH: THE MODERN PERIOD 31
tion period is still quite uncertain. The spectroscopic binary
stars, on the other hand, are, in general, systems of relatively
small dimensions, the revolution periods ranging from five or
six hours, as a minimum, to a few years. The masses of the
systems being assumed to be of the same order, the smaller
the dimensions, the greater the orbital velocity, and the greater
the probability of the detection of the system by means of the
spectrograph, for the amount of the displacement of the lines in
the spectrum is a function of the radial velocity of the star.
Now, if the revolution of a system is accomplished in, say,
two or three days, or even in two or three weeks, or months,
it is possible for an observer to secure ample data for the computa
tion of its definite orbit in a single season. Indeed, if the spectro
graph is devoted to this purpose exclusively and the percentage
of clear nights is large, a single telescope may in one season secure
the data for the orbits of twenty or more systems.
As in the work of their discovery, so in the computation
of the orbits of the spectroscopic binary stars the American
observatories are taking the lead. Inspection of the table
of orbits given in the appendix to this volume will show that
more than 90 per cent of the orbits have been computed' by
astronomers at the Dominion Astrophysical Observatory, at
Victoria, and the Dominion Observatory at Ottawa, in Canada,
and by those at the Lick, Allegheny, Yerkes, Mount Wilson,
and Detroit Observatories in the United States.
While the spectroscopic binary stars have been receiving ever
increasing attention in recent years, the visual binary stars are
by no means being neglected. The work of measuring 1 and
remeasuring the double stars discovered has been carried on
enthusiastically by scores of able observers with small tele
scopes and with large ones. It is impossible to comment upon
all of these or to give details of the hundreds of series of meas
ures they have published but I cannot refrain from referring
here to two of the most prominent observers of the generation
that has just passed away — G. V. Schiaparelli and Asaph Hall.
Schiaparelli's measures are published in two quarto volumes,
the first containing the measures made at Milan with the 8in.
refractor, in the years 1875 to 1885; the second, the series made
with the 18in. refractor at the same observatory in the interval
from 1886 to 1900. Hall's work, carried out with the 26in.
refractor of the United States Naval Observatory at Washington,
32 THE BINARY STARS
is also printed in two quarto volumes, the first containing the
measures made in the years 1875 to 1880; the second, those made
from 1880 to 1891. The working lists of both observers were
drawn principally from the Dorpat and Pulkowa catalogues,
but include many of Burnham's discoveries and some made by
Hough and by others. The high accuracy of their measures
and the fact that they — and Schiaparelli in particular — repeated
the measures of the more interesting stars year after year makes
the work of these observers of the greatest importance.
At present, double stars are regularly measured at two or
three of the larger observatories of this country, at several
important observatories in England and on the continent of
Europe,* and by many enthusiastic amateurs in this country
and abroad. So voluminous is the literature of the subject
that one who wishes to trace the full record of one of the dou
ble stars discovered by Herschel or by Struve in the original
sources must have access to a large astronomical library. This
condition was recognized many years ago, and as early as
1874 Sir John Herschel's A Catalogue of 10,300 Multiple and
Double Stars, Arranged in Order of R. A. was published as a
Memoir of the Royal Astronomical Society. This catalogue
attempted merely to give a consecutive list of the known double
stars, without measures and did not go far toward meeting
the needs of observers or computers. The first really service
able compendium was that published by Flammarion in 1878,
entitled Catalogue des fitoiles Doubles et Multiples en Mouve
ment relatif certain. The volume aimed to include all pairs
known from the actual measures to be in motion; 819 systems
are listed, each with a fairly complete collection of the published
measures, about 14,000 in all, and notes on the nature of the
motion. For thirty years this book formed a most excellent
guide to observers.
The following year, 1879, A Handbook of Double Stars, pre
pared by Crossley, Gledhill, and Wilson, was published in
London — a work that had a wide circulation and that proved
of the greatest service to students of doublestar astronomy.
It is divided into three parts, the first two giving a general
account of doublestar discoveries and methods of observing and
of orbit computation. The third section contains a "Catalogue
* This statement refers to the northern hemisphere; an account of the
work in the southern hemisphere is given on later pages.
HISTORICAL SKETCH: THE MODERN PERIOD 33
of 1,200 double stars and extensive lists of measures." An
appendix gives a list of the principal papers on double stars.
In 1900, Burnham published a General Catalogue of his
own discoveries containing a complete collection of all known
measures of these stars with notes discussing the motion when
such was apparent, and references to the original sources from
which the measures were taken. This proved to be the first
of a series of such volumes. Hussey's catalogue of the Otto
Struve stars, to which reference has already been made, was
published in 1901, and five years later, in 1906, Lewis's great
volume on the Struve stars appeared. This is, in effect, a
revision of the Mensurae Micrometricae and gives all of the
2 stars in the order of their original numbers, disregarding the
inconvenient division into classes. Such of the S "rejected"
stars as have been measured by later observers are also included,
and all or nearly all of the published measures of each pair.
The notes give an analysis and discussion of the motions that
have been observed, and form one of the most valuable features
of the work, for the author had devoted many years to a com
prehensive study of double star astronomy in all its phases.
In 1907, Eric Doolittle published a catalogue of the Hough
stars, all of which he had himself reobserved, and in 1915,
Fox included in Vol. I of The Annals of the Dearborn Observatory
catalogues of the discoveries of Holden and of Kustner with
a new series of measures of these stars. More recently Van
Biesbroeck (Publications of the Yerkes Observatory, Vol. V,
Part I, 1927) has published measures of all of Hussey's pairs,
1,298 in all, that were within reach of observation from the
Yerkes Observatory. Thus all of the longer catalogues of new
double stars discovered at observatories in the northern hemi
sphere, except my own and some other of the very recent ones
and those of Sir John Herschel, have now been revised and
brought up to date, for Sir William Herschel's discoveries,
except the very wide pairs, are practically all included in the
Mensurae Micrometricae.
Every one of the volumes named is most convenient for
reference and contains information not easily to be found else
where; but they were all surpassed by Burnham's comprehensive
and indispensable work, A General Catalogue of Double Stars
vnthin 121° of the North Pole, which was published by The
Carnegie Institution of Washington in 1906. This monu
34 THE BINARY STARS
mental work consists of two parts, printed in separate quarto
volumes. Part I contains a catalogue of 13,665 double stars,
including essentially every pair, close or wide, within the sky
area named, that had been listed as a double star before 1906.
The positions, for 1880, are given, with the discovery date and
measure or estimate. Part II contains measures, notes and
complete references to all published papers relating to each
pair. This catalogue proved to be a most valuable guide to
double star observers and it was in no small measure responsible
for the great activity in double star discovery and observation
in the following years.
In 1917, M. Robert Jonckheere published, in the Memoirs
of the Royal Astronomical Society (Vol. 61) a Catalogue and
Measures of Double Stars discovered visually from 1905 to 1916
within 105° of the North Pole and under 5" Separation. This is,
in effect, an extension of Burnham's General Catalogue, though
the author excluded pairs wider than 5" instead of recording
every pair announced by its discoverer as double and adopted
— 15° instead of —31° or the southern sky limit. The volume
is particularly valuable because it gives in collected form Jonck
heere's own discoveries with measures at a second epoch as
well as at the time of discovery. The other long lists in the
volume are Espin's discoveries and those made at the Lick
Observatory; in all, there are 3,950 entries.
On his retirement from the Yerkes Observatory, Burnham
turned over to Prof. Eric Doolittle, Director of the Flower
Observatory, the material he had accumulated for a revision
or extension of his catalogue. Doolittle set up a card catalogue
planned to contain a complete record of measures and orbits
published after 1906. This catalogue and a collection of books
and pamphlets on double stars came to me after Doolittle's
untimely death in 1920, for, in 1919, in response to his urgent
request, I had promised to carry on the work if he did not live
to complete it.
The result was the publication, early in 1932, of a New General
Catalogue of Double Stars within 120° of the North Pole.*
Designed, in a general way, to supplement Burnham's great work,
it gives for each pair listed in it the earliest available measure
* New General Catalogue of Double Stars within 120° of the North Pole,
by Robert Grant Aitken, in succession to the late Eric Doolittle, Carnegie
Institution of Washington, Publication 417, 2 Vols., 1932.
HISTORICAL SKETCH: THE MODERN PERIOD 35
and all later measures, except those quoted or referred to by
Burnham, together with appropriate notes.
Not all pairs listed in the earlier work, however, are included
in this new catalogue. It was thought that the wider and fainter
pairs could to advantage be omitted, and limits were therefore
set, based upon the apparent magnitude and the angular separa
tion of the components. These limits are defined by the equation
log p = 2.8  0.2m,
in which p is the angular separation and m the apparent magni
tude. The constant, 2.8, sets the limit at 10" for a pair whose
apparent magnitude is 9.0. Although approximately three
out of every ten pairs listed by Burnham are excluded by these
limits, so numerous have been the later discoveries that the new
catalogue has 17,181 entries as compared with Burnham's
13,665. The catalogue includes all measures known to me that
were made prior to 1927.0. A card catalogue of measures
published later is kept up to date at the Lick Observatory.
It has been convenient in this narrative to confine attention
to this point to the double star work done at observatories in
the northern hemisphere, for, until quite recent years it was
there that this branch of astronomy received most attention.
Now, however, conditions are changed and the most active and
fruitful work in the discovery and measurement of double stars is
that carried on at the observatories in South Africa.
We have seen that two of the earliest double stars discov
ered — a Centauri and a Cruris — were stars not visible from
European latitudes; but the first extensive list of double stars
collected at a southern observatory was James Dunlop's cata
logue of 253 pairs observed at Parametta, N.S.W., in the years
1825 to 1827 with a 9ft. reflecting telescope. These stars,
however, are as a rule very wide pairs and are of comparatively
little interest. A few double stars are contained in Brisbane's
Parametta catalogue, published in 1835, and more in the later
meridian catalogues of the Royal Observatory at the Cape of
Good Hope, the Argentine National Observatory at Cordoba,
and of other southern observatories.
The most important early paper on southern double stars
is beyond question the chapter upon them in Sir John Herschel's
36 THE BINARY STARS
Results of Astronomical Observations made during the Years
1834, 1835, 1836, 1837, 1838 at the Cape of Good Hope which
was published in 1847. Innes says, "The sections on double
stars in this work are to the southern heavens what Struve's
Mensurae Micrometricae are to the northern heavens." A
catalogue is given of the discoveries made at Feldhausen, C. G. H.,
with the 20ft. reflector which contains the pairs h3347 to
h 5449, together with measures of such previously known
pairs as were encountered in the "sweeps." Many of the new
pairs are wide and faint, resembling the h stars discovered
at Slough, in England; but many others are comparatively
close, many are very bright, and a number are among the finest
double stars in the southern sky. Another division of this
chapter gives the micrometer measures, made with the fiveinch
refractor, of many of these new pairs and of some of the known
ones. Innes says that "the angles of the pairs are all through
of high excellence"; but Herschel himself points out the sources
of weakness in his measures of distances.
Herschel's station at Feldhausen was not a permanent obser
vatory, and when he returned to England work there was
discontinued; nor was double star work seriously pursued at
any other southern station until about 40 years later. In
1882, a list of 350 new pairs was published by H. C. Russell,
director of the Sydney Observatory, N.S.W., the measures
being made by Russell and by L. Hargrave. In 1884, an addi
tional list of 130 pairs, mostly wide, was published, and in
the following years several lists of measures by these observers
and their colleague, J. A. Pollock, a few of the measured pairs
being new. In 1893, R. P. Sellors published a short list (14 pairs,
all under 2") discovered by him at the same observatory, and
in the following years he contributed many measures of known
pairs and discoveries of a few additional new ones.
A new chapter in the history of double star astronomy in the
southern hemisphere was opened in 1896, when Dr. R. T. A. Innes
joined the staff of the Royal Observatory at the Cape of Good
Hope. Innes had already published as "probably new" two
short lists of 26 and 16 stars, respectively, discovered at Sydney,
N.S.W., in 1895 and 1896, with a 6>iin. refractor and a small
reflector. At the Cape Observatory, in addition to work in
other lines, he continued his double star observing with the
7in., and later with the 18in. (McClean) refractor. With these
HISTORICAL SKETCH: THE MODERN PERIOD 37
instruments he brought the total of his discoveries to 432 and
made fine series of measures of these and of other southern
double stars. In 1903 he became Government Astronomer at
the Union Observatory, Johannesburg, South Africa. Here
he worked with a 9in. refractor until April, 1925, when the
26J^in. Grubb refractor, ordered some years before the war
began, was finally installed.
In August, 1925, Dr. W. H. van den Bos, who had been doing
excellent double star work at Leiden, came to the Union Obser
vatory. A plan, which Innes had long cherished, for a systematic
survey of the southern sky along the general lines of the one
of the northern sky carried out at the Lick Observatory, was
immediately put into execution. The major part of this survey
has been carried out by van den Bos, though Dr. W. S. Finsen
and other assistants in the observatory (as well as Innes himself
until he retired in 1927) have participated. The survey is still
in progress but is nearing completion. Including the earlier
work at the Union Observatory by Innes and others, fully
4,000 double stars had been discovered before the close of the
year 1931. Innes' own discoveries total 1,613, Finsen's 300,
while those of van den Bos exceed 2,000. These are all close
pairs, comparable in every respect with those discovered at the
Lick Observatory.
In 1911, Hussey accepted the directorship of the observatory
of the La Plata University, Argentina, in addition to his duties
at Ann Arbor, Michigan. During his periods of residence at
La Plata he used the 17in. refractor in searching for and measur
ing southern double stars. His discoveries there brought
his total number of new pairs up to 1,650, and measures of these
later pairs were promptly published.
Mr. Bernhard H. Dawson assisted Prof. Hussey at La Plata
until 1917, when Hussey resigned the directorship, and has
since continued to give part of his time to double star work.
Hussey now found it possible to plan for further double star
work in the southern hemisphere with a more powerful telescope,
the funds for the construction of which had been provided by
his friend and college classmate, Mr. R. P. Lamont. A 27in.
lens was ordered from the John A. Brashear Company, and
Hussey personally supervised the designs for the mounting.
Delays were encountered, however, and the war came on, with
38 THE BINARY STARS
the result that the telescope was not ready for use until the
summer of 1926.
Professor Hussey, in 1924, had personally selected a site at
Bloemfontein, South Africa, for his southern station, and was
on his way there to supervise the erection of the telescope and to
carry on double star work when he died of heart disease, in Lon
don, on Oct. 23, 1926. The telescope, however, was erected in
accordance with his plans, Prof. R. A. Rossiter being placed in
charge, with Morris K. Jessup and Henry F. Donner as assistants.
Arrangements were made with the Union Observatory for
cooperative work in prosecuting the survey of the southern
heavens, with gratifying results. By October, 1931, 4,712 new
double stars were discovered and measured on one or more
nights; 1,961 by Rossiter, 1,424 by Jessup, and 1,327 by Donner.
These measures, for the most part, are as yet unpublished.*
At the Union Observatory, all stars as bright as 9.0 magnitude
in the Cape Photographic Durchmusterung are examined, as well
as those of fainter photographic magnitude which are estimated
to be as bright as 9.0 visual magnitude. The distance limit for
pairs listed as double stars is set by the curve log p = 2.5 — 0.2m,
which gives 5"0 for a pair of 9.0 magnitude.
At the Lamont Hussey Observatory, the survey is being carried
to all stars as bright as 9.5 in CPD, and the distance limit is
that given by the curve log p = 2.625 — 0.2m, giving 6"75 at
9.0 magnitude, and pairs even wider (to the limit set by
log p = 2.8 — 0.2m) are retained. The result is that while
3,206 pairs fall within the limits of the Union Observatory curve,
the percentage of pairs fainter than 9.0 and comparatively wide
is very high.
Any statistical discussion of the number and distribution of
double stars based upon the material that will be available after
these southern surveys are completed, must take account of
the systematic difference between visual and photographic
magnitudes and of those between the various systems of visual
magnitudes. To be significant, such a discussion must include
* Rossiter's first list of measures of the pairs of his own discovery has since
been published in the Memoirs of the Royal Astronomical Society, Vol. LXV,
Part II, 1933. In his introductory note he states that his discoveries to
date exceed 2,350 pairs. Measures only of those found prior to October,
1932, however, a total of 2,232 pairs, are included in his paper. The list
is notable for the large number of very close faint pairs contained in it.
HISTORICAL SKETCH: THE MODERN PERIOD 39
the double stars in both hemispheres and should be based upon
accurate photometric magnitudes.
Innes, in 1899, published a Reference Catalogue of Southern
Double Stars, designed to include "all known double stars having
southern declination at the equinox of 1900." The author,
however, set limits denning the pairs of stars to be regarded
as "double stars," the limits ranging from 1" for pairs of the
ninth magnitude to 30" for those of the first magnitude. In
principle, this procedure is correct, but Innes* actual narrow
limits are open to criticism. He abandoned these limits when,
with the assistance of Dawson and van den Bos, he compiled
his looseleaf catalogue that was planned to contain every
known double star within the limits of the Cape Photographic
Durchmusterung {i.e., south of —19° at the equinox of 1875.0)
"that had been measured on more than one occasion as well as
many measured only once." This looseleaf catalogue was
completed in 1927, but it was regarded as merely a temporary
guide to southern observers, the declared intention being to
issue a complete catalogue in more permanent form after the
termination of the survey of the southern skies, initiated in 1925.
Meanwhile, a card catalogue of all discoveries and published
measures is kept up to date at the Union Observatory.
Our knowledge of the spectroscopic binary stars in the far
southern skies rests almost entirely upon the work carried on
at the D. O. Mills Station of the Lick Observatory, established
at Santiago, Chile, in 1903, and maintained until April, 1929,
when it was sold to the Catholic University of Chile. The
instrumental equipment consisted of a 37^in. silveronglass
reflector and spectrographs similar in design to those in use on
Mount Hamilton. The working program was the measurement
of the radial velocities of the stars and nebulae which are too
far south to be photographed at the Lick Observatory itself.
The discovery of binary stars was not the object in view, but a
large percentage of the entire number of these systems known
at the present time were found at this Station in the 26 years
of its existence. When we add to this number the spectroscopic
binary stars with southern declinations which have been detected
by observers at stations in the northern hemisphere, we shall
find that in this field there is little or no disparity between the
two hemispheres of the sky.
40 THE BINARY STARS
References
In addition to general accounts of the binary stars in standard textbooks
on astronomy, reference may be made to two recent publications:
Henkoteatj, F. C. : "Double and Multiple Stars," Handbuch der Astrophysik,
Band VI, Chap. 4, pp. 299474, Berlin, 1928.
Baize, P.: L'Astronomie des fitoiles Doubles, Bull. Soc. Astron. de France
44, 268, 359, 395, 505; 46, 21, Paris, 19301931.
[Dr. Hamilton M. Jeffers and Mrs. Frances Greeby at Lick Observatory have
been keeping up visual binary orbit data. Their new general catalogue is now
ready for publication. It lists about 65,000 double stars, and indicates each
pair for which an orbit has been calculated. North of —20°, the compilation
has been made at Lick Observatory. South of —20°, the catalogue of Dr.
Willem H. van den Bos is incorporated. Mrs. Greeby plans to keep up the
catalogue of observations, and additions and changes to the general catalogue.
Attention should also be called to the new catalogue (to be published) by
Charles Worley of the Naval Observatory. — J.T.K.]
CHAPTER III
OBSERVING METHODS, VISUAL BINARY STARS
The operation of measuring a double star is a very simple
one. The object is to define at a given instant the position of
one star, called the companion, with respect to the other, known
as the primary. When the two stars are of unequal magnitude
the brighter is chosen as the primary; when they are of equal
brightness, it is customary to accept the discoverer's designations.
From the first work by Sir William Herschel, the measures
have been made in polar coordinates; and since about 1828,
when Sir John Herschel recommended the practice, the posi
tion angle has been referred to the North Pole as zero point
and has been counted through 360°.
That is, the position angle is the angle at the primary star
between the line drawn from it to the North Pole and one
drawn from it to the companion, the angle increasing from
zero when the companion is directly north, through 90° when
it is at the east, 180° when it is south, 270° when it is west,
up to 360° when it is once more directly north. The distance
is the angular separation between the two stars measured at right
angles to the line joining their centers. The two coordinates
are usually designated by the Greek letters and p, or by the
English letters p and s.
THE MICROMETER
The filar or parallelwire micrometer is the instrument now in
almost universal use for visual measurements of double stars.*
* Mr. F. J. Hargreaves has recently perfected a comparisonimage microm
eter which he finds more accurate than the filar micrometer, particularly
in the measurement of angular distances. Two artificial star images are
projected into the field of view side by side with the images of the double
star to be measured. The artificial star images may be made comparable
to those of the real stars in both color and brightness and their position
angle and angular separation may be brought into accurate agreement
with those of the double star. A full description is given in the Monthly
Notices, R.A.S. (92, 72, 1931), but the instrument has not yet come into
general use.
41
42 THE BINARY STARS
A complete description of it is not necessary here; for this, the
reader is referred to Gill's article on the Micrometer in the
Encyclopaedia Britanniea (9th ed,), in which other forms are also
described. Essentially it consists of a tube or adapter firmly
fitted into the eye end of the telescope and carrying on its outer
end a graduated circle (the position circle) reading from 0° to
360° in a direction contrary to the figures on a clock dial. A
circular plate fitting closely within the position circle and
adjusted to turn freely within it carries an index, or a vernier,
Plate II. — The micrometer for the 30in. refractor, Lick Observatory.
or both, to give the circle reading. In the micrometers in
use at the Lick Observatory, this plate is rotated about the
optical axis of the telescope by an arm carrying a pinion which
meshes into rack teeth cut on the outer circumference of the
position circle, A clamp is provided to hold the plate and
circle together at any desired reading, and a tangent screw
to give a slow motion. Upon the vernier plate an oblong box
is mounted within which the parallel wires or threads (they
are usually spider lines) are placed. This box is movable
longitudinally by a wellcut, but not very fine screw. One
thread, the fixed thread, is attached to the inner side of the
upper plate of the box, and the other, the micrometer or mov
able thread, is attached to a frame or fork which slides freely
OBSERVING METHODS, VISUAL BINARY STARS 43
in the box longitudinally, but without any lateral play. The
fork is moved by a very fine and accurately cut screw which
enters the box at one end. At its outer extremity, this screw
carries a milled head divided into 100 parts, the readings increas
ing as the screw draws the micrometer thread toward the head.
Strong springs at the opposite end of the fork carrying this
thread prevent slack or lost motion.
The two threads, the fixed and the micrometer, must be so
nearly in the same plane — the focal plane of the objective —
that they can be brought into sharp focus simultaneously in an
eyepiece of any power that may be used, but at the same time
must pass each other freely, without the slightest interference.
Instead of a single .fixed thread, some micrometers carry sys
tems of two, three, or more fixed threads, and frequently also
one or more fixed transverse threads. Some also substitute
two parallel threads separated a few seconds of arc for the
single movable thread. For double star work, the simple
micrometer with only two threads is unquestionably to be
preferred, and even for comet, asteroid, satellite, and other
forms of micrometric work I regard it as superior to the more
complicated forms and less liable to lead to mistakes of record.
Not all observers, however, will agree with me on this last point.
The telescope is assumed to be mounted stably and to be in
good adjustment. Assured as to these two points and as to the
firm attachment of the micrometer to the telescope tube so that
the zero reading of his position circle shall remain constant,
the double star observer has still to determine the value of one
revolution of his micrometer screw and the zero or north point
reading of his position circle before beginning actual measure
ments. The reading for coincidence of the threads is elimi
nated by the method of doubledistance measures, as will be
shown presently, and the distances themselves are, in general,
so small, and modern screws so accurate, that irregularities
in the screw and corrections for temperature may be regarded
as negligible. If desired, however, they may be determined
in connection with measures for the revolution value.
THE ZERO POINT
The determination of the zero point will be considered first.
The simplest practical method, and the one adopted by observers
generally, is to put on the lowest power eyepiece that utilizes
44 THE BINARY STARS
the entire beam of light, direct the telescope upon an equatorial
star near the meridian, stop the driving clock, and turn the
micrometer by the box screw and the positioncircle pinion
until the star "trails" along the thread across the entire field
of view. The star should be bright enough to be seen easily
behind the thread, but not too bright. With the 12in. telescope
I find a star of the seventh or eighth magnitude most satisfactory;
with the 36in. telescope, one of the ninth or tenth magnitude.
A little practice will enable the observer to determine his "par
allel" reading with an uncertainty not greater than onefifth
of one division of his circle. On the micrometer used with the
36in. telescope, this amounts to 005. Several independent
determinations should be made. If the micrometer is not
removed from the telescope and is set firmly to the tube, it is
probable that the parallel reading need be checked only once or
twice a week. When, as at the Lick Observatory, the micrometer
is liable to be removed almost any day and is certainly removed
several times every week, the observer very promptly forms the
habit of determining the parallel at the beginning of his work
every night; my own practice is to check the value at the close
of work also.
Ninety degrees added to the parallel gives the north point or zero
reading.
REVOLUTION OF THE MICROMETER SCREW
The value of one revolution of the micrometer screw should
be determined with the greatest care and the investigation
should be repeated after a reasonable time interval to detect
any wear of the screw. Two different methods of procedure
are about equally favored by observers: the method of transits
of circumpolar stars and the method of direct measures of the
difference in declination of suitable pairs of stars.
In the first method the position circle is set for the zero reading
(i.e., 90° from the reading for parallel) and the telescope turned
upon the star a short time before it culminates. (The driving
clock, of course, is stopped.) Set the micrometer thread just
in advance of the star as it enters the field of view (it is convenient
to start with the milled head set at zero of a revolution) and note
the time of the star's transit either on the chronograph or by
the eyeandear method. Advance the thread one revolution
or a suitable fraction of a revolution and take another transit,
OBSERVING METHODS, VISUAL BINARY STARS 45
and repeat this procedure until the star has crossed the entire
field of view. A lowpower eyepiece should be used and the
series of measures so planned that they will extend over from
forty to eighty revolutions of the screw, about half of the transits
being taken before the star crosses the
meridian, the other half after. Great
care must be taken not to disturb the
instrument during the course of the
observations for the slightest changes
in its position will introduce errors
into the measures. It is well to re
peat the observations on a number of
nights, setting the telescope alternately
east and west of the pier. A sidereal
timepiece should be used in recording
the time of transits and if it has a large
rate, it will be necessary to take this
into account.
In Fig. 2, let P be the pole, EP the observer's meridian,
ab the diurnal path of a star, AS the position of the micrometer
thread when at the center of the field and parallel to an hour
circle PM, and BS' any other position of the thread. Now
let Wo be the micrometer reading, to the hour angle, and To
the sidereal time when the star is at S, and m, t, and T the cor
responding quantities when the star is at S', and let R be the
value of one revolution of the screw.
Through S' pass an arc of a great circle S'C perpendicular to
AS. Then, in the triangle CS'P, rightangled at C, we have
CS' = (m  m )R, S'P = 90°  5, CPS' = t  to = T  T
Fig. 2.
and we can write
sin [(w — m )R] = sin (T — T ) cos 8
or, since (wi — m^)R is always small,
cos 6
(m  m )R = sin (T  To) rjry
Sill X
Similarly, for another observation,
{m'  m )R = sin (T'  To)
cos 8
shTT 7
(1)
(2)
(3)
46 THE BINARY STARS
Combining these to eliminate the zero point,
(m'  m)R = sin {T  T ) ^p  sin (T  To) ^p (4)
from which the value of R is obtained. The micrometer readings
are supposed to increase with the time.*
If 80 transits have been taken, it will be most convenient
to combine the first and the fortyfirst, the second and the
fortysecond, and so on, and thus set up 40 equations of condi
tion of the form of Eq. (4). The solution of these equations by
the method of least squares will give the most probable value
for R. The value of R given by Eq. (4) must be corrected for
refraction. It will suffice to use the approximate formula
dR = R tan 1" cot (5  <p)r (5)
where r is the mean refraction, 5, the declination of the star, and <p,
the latitude of the observer. If a star is observed at lower cul
mination, 5 must be replaced by (180° — 8).
If the value of R is to be detennined by direct measures of
the difference of declination between two stars, the stars should
satisfy the following conditions: they should lie on, or very
nearly on, the same hour circle; their proper motions as well
as their absolute positions at a given epoch should be accurately
known; they should be nearly of the same magnitude and,
if possible, of nearly the same color; the difference of declina
tion should amount to from fifty to one hundred revolutions
of the micrometer screw; and, since this will ordinarily exceed
the diameter of the field of view of the eyepiece, one or more
intermediate stars (whose positions do not need to be so accu
rately known) should lie nearly on the line joining them and at
convenient intervals to serve as steps.
There are not many pairs of stars which answer all of the
requirements. Probably the most available ones are to be
found in the Pleiades and other open clusters which have been
triangulated by heliometer observations.
The measures should be made only on the most favorable
nights and at times when the stars are high enough in the sky
to make the correction for refraction small. The difference
of declination should be measured from north star to south
star and also in the opposite direction and the measures should
* From Campbell's Practical Astronomy.
OBSERVING METHODS, VISUAL BINARY STARS 47
be repeated on several nights. If extreme accuracy is desired
in the refraction corrections the thermometer and barometer
should be read at the beginning and also at the end of each
set of measures, and if the effect of temperature is to be included
in the determination of R, measures must be made at as wide a
range of temperature as is practicable.
In making the reductions, the star places are first brought
forward from the catalogue epoch to the date of the actual
observations by correcting rigorously for precession, proper
motion, and the reduction from mean to apparent place. The
apparent place of each star must then be corrected for refrac
tion. It will generally be sufficiently accurate to use Com
stock's formula, in the following form :
Refraction in 8 = — ... , . tan z cos q (6)
4oU + 1
where z is the apparent zenith distance, and q the parallactic
angle of the star, b the barometer reading in inches and t the
temperature of the atmosphere in degrees Fahrenheit. In
practice I have found it more convenient to correct each star
for refraction in the manner described than to correct the differ
ence in declination by the use of differential formula.
The following pairs of stars in the Pleiades have actually
been used by Prof. Barnard in determining the value of one
revolution of the micrometer screw of the 40in. telescope of
the Yerkes Observatory:
BD Mag. BD Mag. AS
+ 23°537 (7.5) and + 23°542 (8.2) 696"l9
+ 23°516(4.8) and + 23°513 (9.0) 285.94
+ 23°557(4.0) and + 23°559 (8.4) 599.58
+ 23°561 (7 . 5) and + 23°562 (7.8) 479 . 1 1
+ 23°558(6.2) and + 23°562 (7.8) 401.10
+ 23°563 (7 . 2) and + 23°569 (7 .5) 494 . 14
+ 23°557(4.0) and +23°558(6.2) 300.25
+ 23°507(4.7) and + 23°505(6.5) 633.40
The differences in declination given in the final column are
for the epoch 1903.0 and are the results of Dr. Elkin's measures
with the Yale heliometer.
The last pair in the list consists of the bright stars Electro, and
Celaeno, and the table that follows gives the measures of them
made by Barnard, in 1912, to determine the screw value for the
48
THE BINARY STARS
o
08 "
CQ
0>
bC
£ 5
3 d
.2 .1
II
a
cn
&q
HQIQNQOIQ
CO CO CO CO CO
88
0000000
I I I I I
M CO§« N S §
m cm cm' cm e* ei cm
(NNNNHNN
0000000
b' ©©'©©©©
+++++++
g
o o >o o o o o
eo c5 cm ^ ih •*
"Si C* CM ih H w CM
+++++++
SN 11 CO CO CO W3
10 eo ■<* c« eo (N
t b t^ t~ t» t t^
<N C* CM <N CM CM C*
CO iH CO
£2 <5 «5
co eo us
00 »© © cm •* co ^
OOSHtON Hft
CM rt <N CN 0* C* iH
00 GO GO 00 00 00 00
O
©
+
43
■s
1— I
os
iH
§
e* a
0Q
OBSERVING METHODS, VISUAL BINARY STARS 49
micrometer of the 40in. Yerkes refractor. Step stars of magni
tude 11.0 and 11.5, respectively, lying nearly in the line joining
the two bright stars were used to reduce the intervals actually
measured. Both the tube of the 40in. telescope and the screw
of the micrometer are of steel and therefore mutually correct
each other in temperature changes, at least approximately;
but the focal length of the object glass is % in. shorter in winter
than in summer whereas the tube shortens only 3^ in. A
slight correction is therefore necessary if all of the measures
are to be reduced to the focus for a common temperature.
The column Scale reading, in the table of measures, gives the
readings for focus on the draw tube of the telescope and the
following column, the corrections required to reduce the meas
ures to the focal length corresponding to a temperature of
60°F. The remaining columns are selfexplanatory.
If a suitable measuring engine, like those used in measuring
positions of images on a photographic plate, is available, it may
be used to measure the value of the revolution of the micrometer
screw in terms of millimeters, provided the pitch of the screw
is known accurately. Dividing the result by the focal length
of the telescope, in millimeters, and multiplying by the value
of the radius of a circle expressed in seconds (206,264.8) will
reduce it to seconds of arc. Burnham's original measure of the
screw value of the 36in. micrometer of the Lick Observatory
was #'907 ± 0"006. This was based on measures of the differ
ence in declination of two stars, seven different pairs being meas
ured. Wright has recently measured the screw under one of
our measuring engines, finding 9"9045 ± 0'/0005, the value
corresponding to a temperature of 62° F.
MEASURING A DOUBLE STAR
When the telescope has been directed upon the star and
clamped, the star is brought up to the threads by means of the
screw moving the entire micrometer box. The position angle
is then measured, and in doing this my practice is to run the
micrometer thread well to one side of the field of view, bring
the double star up to the fixed thread by means of the screw
moving the box and then rotate the micrometer by means of
the pinion provided, keeping, meanwhile, the fixed thread
upon the primary star, until the thread also passes centrally
50 THE BINARY STARS
through the companion star. It is most convenient to manipu
late the box screw with the left hand and the pinion with the
right.
The tangent screw giving a slow motion in position angle
is never used; in fact, it has been removed from the micrometer.
When the seeing is good, the star images round, small, and
steady, it is easy to hold both images on the thread until the
eye is assured of their precise bisection. Under less favorable
conditions a rapid to and fro motion of the box screw places
the stars alternately on either side of the thread while the
pinion is being rotated backward and forward until the eye
is satisfied of the parallelism of the thread to the line joining the
centers of the star images.
Ordinarily four independent settings for position angle are
made, the circle being read, not by the vernier but by an index,
directly to half degrees in the case of the 12in. micrometer,
to quarter degrees in the case of the 36in., and by estimation
to the onefifth of a division, i.e., to 0?1 and 0°05, respectively.
To insure the independence of the readings, the micrometer is
rotated backward and forward through an arc of 60° to 80°
after each setting. The eye is, of course, removed from the
eyepiece, and the box is turned directly with the hands, without
the use of the rotation pinions.
The circle is next set to a reading 90° greater (or less) than
the mean of the readings for position angle and the distance
is measured by bisecting one star with the fixed thread, the
other with the micrometer thread. It is most convenient to
turn the micrometer screw with the right hand, the box screw
with the left. Then interchange the threads, placing the
micrometer thread on the first star, the fixed thread on the
other. The difference between the two readings of the microm
eter screwhead gives the double distance, i.e., twice the
angular separation, and eliminates the zero or coincidence
reading. Three measures of the double distance are generally
made. The milled head of the screw, which is divided to
hundredths of a revolution, is read to the 1/1,000 revolution by
estimation. Care is always taken to run the micrometer thread
back several seconds of arc after each setting and to make
the final turn of the screw at each bisection forward or against
the springs. The bisection of the star by the fixed thread should
be made anew at each setting with the micrometer screw, because,
OBSERVING METHODS, VISUAL BINARY STARS 51
under even the best conditions, it cannot be assumed that the
star images will remain motionless during the time of observation.
Any ordinary notebook will answer as a record book. At
the Lick Observatory, we have found convenient a book 7 by
8% in. containing 150 pages of horizontally ruled, sized paper
suitable for ink as well as pencil marks. The observing record
is made with pencil, the reductions in ink. No printed forms
are necessary or even convenient. A sample entry taken from
my observing book shows the form of record adopted, and also
the very simple reductions:
36" Sat. Jan. 27, 1917
80Tauri = 2 554
Parallel = 10?25
128?75re/0
9 ±
49.401 9.581
4^3
129.70 Am
= 3
9.400 9.578
1,000
129.30
9.403 9.580
2to2 +
130.40
Well separated with
49.401 9.580
9.401
520power
129.54
100.25
2)0.179
29.°3 = O 0.089# = 88 = p
Two or three such entries are ordinarily made to the page. The
column at the left records the four settings for position angle;
the mean is taken and the reading of the circle for parallel
plus 90° is subtracted to obtain the position angle. Whether
this value is the correct one or whether 180° is to be added to
it is decided by the note made of the quadrant while observing
— nf in the present case. When recording the quadrant, which
is done after the position angle settings have been entered,
I record also an estimate of the distance and of the difference of
magnitude of the components, or, when the companion is very
faint, a direct estimate of its magnitude. At this time, too, I
record, at the right, the date, the sidereal time to the tenth of
an hour, the power of the eyepiece used, an estimate of the seeing
on a scale on which 5 stands for perfect conditions and any
observing notes» Measures of distance are then made and
recorded. Here the reduction consists in taking half the differ
ence of the two means and multiplying the result by the value
of one revolution of the micrometer screw (in this instance
9 '907).
52 THE BINARY STARS
The results are transferee! to a "ledger," or, preferably, to
the cards of a card catalogue, the date being recorded as a decimal
of the year. The ledger entry for the above observation is :
80 Tauri = 2554.
1917.075 29°3 0'/88, Am = 3, 4^3, 1000, 2 to 2 + , bk. 87,147
the last item being the number and page of the observing book.
Practically all observers agree in the method of measuring
the angular distance, but many prefer a somewhat different
procedure for determining the position angle. They bring the
two threads fairly close together — to a separation twice or
three times the diameter of the primary's apparent disk —
and then, placing the two stars between the threads, turn the
micrometer until the line joining the stars appears to be parallel
to the threads. I have found that I can secure equally satis
factory measures by this method when the two stars are well
separated and of nearly equal magnitude, but not when the
angular distance is small or when the stars differ much in bright
ness. While it may be a matter of personal adaptation I am
inclined to think that measures made in this manner are more
likely to be affected by systematic errors than those made by
the method first described.*
Whatever method is adopted, it is of the first importance
that the head of the observer be so held that the line between
his eyes is parallel or perpendicular to the line joining the two
stars. I can make the bisections with more assurance when
the line between the eyes is parallel to the one joining the two
stars, and hold my head accordingly unless the line is inclined
more than 45° to the horizon. Some observers prefer the
perpendicular position. When the line joining the stars makes
an angle of approximately 45° to the horizon, it is well to make
settings in both positions of the eyes. In pairs with components
of unequal magnitude, a systematic difference between the two
sets of readings may be expected.
There are some other precautions that must be taken to secure
satisfactory results. The star images as well as the threads
must be brought sharply into focus; the images must be sym
* It should be noted that one or two good observers determine the position
angle by setting the threads as nearly as possible perpendicular to the line
joining the two stars. This practice is not recommended.
OBSERVING METHODS, VISUAL BINARY STARS 53
metrically placed with respect to the optical axis; and the threads
must be uniformly illuminated on either side. In modern
micrometers the ulumination is usually provided by a small
incandescent lamp placed in such a position that a small mirror
can throw the light through a narrow opening in one end of the
micrometer box. This mirror can be rotated through 90° thus
permitting a variation in the intensity of the light from full
illumination to zero. Suitable reflectors placed within the
micrometer box, at the opposite end, insure equality in the illumi
nation on both sides of the threads. Glass slides or color filters
can also be placed in front of the opening admitting the light
in order to vary its intensity or its color as may be desired.
The earlier double star observers frequently illuminated the
field of view instead of the threads and an occasional observer
still advocates this practice, but the great majority, I think,
are agreed that this is a less satisfactory arrangement.
It is hardly necessary to say that the micrometer threads
must be stretched to a tension sufficient to keep them perfectly
straight, even when the atmosphere is very moist, and that they
must be free from dust or other irregularities and accurately
parallel. A cocoon of spider thread should be obtained from
an instrument maker and kept on hand with the necessary
adjusting tools and the micrometer threads replaced as often
as they become unsatisfactory. A little practice will enable
the observer to set a thread in position in a very short space
of time; in fact, from Burnham's days to the present time, a
new thread has frequently been set into the 36in. micrometer
during the night and observing resumed within an hour.
The most important precaution to be taken in double star
observing is quite independent of instrumental adjustments.
It is to make measures only on nights when the observing condi
tions are good. Measures made under poor observing condi
tions are at best of little value, and at worst are a positive
nuisance to the student of double star motions. They annoy
or mislead him in his preliminary investigations and are prac
tically rejected in his later work. I make this statement with
all possible emphasis.
It is of almost equal consequence to select stars suited to
the power of the telescope employed. This, however, is to a
considerable extent a matter involving the personal equation.
A Dawes, a Dembowski, or a Burnham can measure with
54
THE BINARY STARS
comparatively small apertures stars that other observers find
difficult with much larger telescopes.
MAGNITUDE ESTIMATES
It is well known that the magnitudes assigned to the com
ponents of the same double star by different observers fre
quently show a range that is excessively large. Whatever excuse
there may have been for this in earlier days, there is certainly
little at the present time when the magnitudes of all of the
brighter stars are given in the photometric catalogues and
those of all stars to at least 9.5 magnitude in the various
Durchmusterungen. It is certainly advisable to take the com
bined magnitude of the two components (or the magnitude of
the brighter star, if the companion is very faint) from these
sources instead of making entirely independent estimates.
The difference of magnitude is then the only quantity the double
star observer need estimate. If this difference is not too
great it can be estimated with comparative accuracy; if one
component is very faint, a direct estimate of its brightness
may be based upon the limiting magnitude visible in the tele
scope used, care being taken to allow for the effect of the bright
companion which will always make the faint star appear fainter
than it really is.
To derive the brightness of each component when the com
bined magnitude and the difference of magnitude are known,
we have the relations, A = C + x, B = A + d, in which A and
B are the magnitudes of the brighter and fainter component,
respectively, C the combined magnitude, and d the estimated
difference of magnitude, while x is given by the equation
x =
)
0.4
We may tabulate x for different values of d as in the follow
ing table which is abbreviated from the one in Innes' Reference
Catalogue:
d
X
d
X
0.0
0.75
1.5
0.25
0.25
0.6
2.0
0.15
0.5
0.5
2.5
0.1
0.75
0.4
3.0
0.05
1.0
0.3
4.0
0.0
OBSERVING METHODS, VISUAL BINARY STARS 55
To illustrate the use of the table let d, the observed difference
in brightness, be 0.7m (it is desirable to estimate the difference
to the nearest tenth of a magnitude), and let the photometric
magnitude, C, be 7.0. Then, from the table, x = 0.4, and the
magnitudes of A and B are 7.4 and 8.1 (to the nearest even tenth).
Conversely, we may find C from A and B.
THE OBSERVING PROGRAM
It has happened in the past that certain wellknown double
stars have been measured and remeasured beyond all reasonable
need, while other systems of equal importance have been almost
entirely neglected. The general catalogues described in the
preceding chapter make it comparatively easy for observers
to avoid such mistakes hereafter. In the light of the knowledge
these catalogues give of past observations and of the motions
in the various systems, the observer who wishes his work to be of
the greatest possible value will select stars which are suited to
his telescope and which are in need of measurement at a given
epoch either because of scarcity of earlier measures or because
the companion is at a critical point in its orbit.
It. has often been said that a careful set of measures of any
pair of stars made at any time is valuable. Granting this to
be so, it is certain that its value is greatly enhanced if it is
made to contribute to the advancement of a program having
a definite end in view. If the aim is to increase the number
of known orbits as rapidly as possible, attention should be
centered upon the closer pairs, particularly those under 0"5
and those which have already been observed over considerable
arcs of their orbits. Though I have felt myself obliged to devote
my observing time in recent years to the remeasurement of as
many as possible of the pairs of my own discovery, I am none
the less convinced that it is, in general, wise for an observer
possessing the necessary telescopic equipment to devote his
energy largely to the measurement of a limited number of
rapidly moving systems, repeating the measures every year,
or every two or five years, as may be required by the rapidity
of the orbital motion, for a long series of years. Such a series
can be investigated for systematic as well as accidental errors
of measure far more effectively than an equal number of meas
ures scattered over a much larger program, and will add more
to our knowledge of the orbits of the binary systems. The
56 THE BINARY STARS
wider pairs, and particularly those in the older catalogues,
now need comparatively little attention, so far as orbital motion
is concerned. Even moderately close pairs, with distance
from 1" to 5", need, in general, to be measured but once in
every 10 or 20 years. Useful programs, however, may be
made from wider pairs for the detection of proper motions,
or for the determination of the relative masses in binary systems
by means of measures connecting one of the components with
one or more distant independent stars. Photographic measures
of these wider pairs are, in general, more accurate than visual
ones.
It is hardly necessary to add that an hour in the dome on
a good night is more valuable than half a dozen hours at the
desk in daylight. Everything possible should therefore be
done to prevent loss of observing time. In this connection I
have found charts based on the Durchmusterung invaluable
for quick identification of stars.
THE RESOLVING POWER OF A TELESCOPE
It has been shown that the diffraction pattern of the image
of a point source of light, like a star, formed by a lens "is a
disk surrounded by bright rings, which are separated by cir
cles at which the intensity vanishes."*
Schuster gives the formula
P=rnf± (7)
in which p is the radius of a circle of zero intensity (dark ring),
D the diameter of the lens, / its local length, \ the wave length
of the light from the point source, and m a coefficient that
must be calculated for each ring. For the first dark ring it is
1.220, and the values for the successive rings increase by very
nearly one unit. Nearly all of the light (0.839) is in the cen
tral disk, and the intensity of the bright diffraction rings falls
off very rapidly. Now it is generally agreed that the mini
mum distance at which a double star can be distinctly seen
as two separate stars is reached when the central disk of the
image of the companion star falls upon the first dark ring of
the image of the primary, and the radius of this ring, expressed
in seconds of arc, is therefore frequently called the limit of the
* Schuster, Theory of Optics, p. 130, 1904.
OBSERVING METHODS, VISUAL BINARY STARS 57
telescope's resolving power. If we adopt for X the wave length
5,500 A, the expression for p in angular measure becomes
A  5r45 (to
P  ~p (8)
from which the resolving power of a telescope of aperture D
(in inches) may be obtained. For the 36in. Lick refractor, the
formula gives 0T15, for the 12in., 0:'45.
It will be observed that the resolving power as thus derived
rests partly upon a theoretical and partly upon an empirical
basis. When the central disk of each star image of a pair falls
upon the first dark ring of the other image, the intensity curve
of the combined image will show two maxima separated by a
distinct minimum. When the disks fall closer together, this
minimum disappears, the image becomes merely elongated,
perhaps with slight notches to mark the position of the dis
appearing minimum. The pair is now no longer "resolved,"
according to the definition given, but to the experienced observer
its character may still be unmistakable. For example, in the
Lick Observatory double star survey, Hussey and I have found
with the 36in. at least five double stars with measured distances
of O''ll or less, the minimum for each observer being 0"09;
and we have found many pairs with the 12in. telescope whose
distances, measured afterward with the 36in., range from 0"20
to 0"25. In all these cases the magnitudes were, of course,
nearly equal.
Lewis* published a very interesting table of the most difficult
double stars measured and discovered by various observers
using telescopes ranging in aperture from 4 to 36 in. He tabu
lated in separate columns the values for the bright and faint
pairs of nearly equal magnitude, and for the bright and faint
pairs of unequal magnitude, each value representing the mean
of about five of the closest pairs for a given observer and telescope.
A final column gave the theoretical resolving power derived, not
from the equation given above, but from Dawes' wellknown
empirical formula — resolving power equal 4' 56 divided by the
aperture in inches (a) — which assumes the two stars to be of
about the sixth magnitude. Lewis found that, in general, this
formula gave values which were too small even for the bright
* The Observatory, 37, 378, 1914.
58 THE BINARY STARS
equal pairs, and he suggested the following as representing more
precisely the results of observation :
4'.'8
Equal bright pairs > mean magnitudes 5.7 and 6.4
8'' 5
Equal faint pairs — — > mean magnitudes 8.5 and 9.1
16'.'5
Unequal pairs — — > mean magnitudes 6.2 and 9.5
36''0
Very unequal pairs — — > mean magnitudes 4.7 and 10.4
Lewis was careful to state that his table did not necessarily
represent the minimum limits that may be reached with a
given telescope under the best conditions, and I have just
shown that it does not represent the limits actually reached
at the Lick Observatory. Taking from each of the three lists
of new double stars 1,026 to 1,274, Hu 1 to Hu 1,327, and
A 1 to A 3104 "about five" of the closest bright, and closest
faint, equal pairs discovered by each of the three observers,
Burnham, Hussey, and Aitken — 29 pairs in all — I find the
following formula for the 36in. telescope:
4 '.'3
Equal bright pairs —^—t mean magnitudes 6.9 and 7.1
6'/l
Equal faint pairs ^— > mean magnitudes 8.8 and 9.0
The most interesting point about these formulas is that
they show much less difference between the values for faint
and bright pairs than Lewis' do.
While it is a matter of decided interest to compare the limits
actually attained with a given telescope with the theoretical
resolving power, an observer, in making out his working pro
gram for double star measurement, will do well to select pairs
that run considerably above such limiting distances. My
deliberate judgment is that, under average good observing con
ditions, the angular separation of the pairs measured should
be nearly double the theoretical limit. Observers with the
most powerful telescopes, however, are confronted with the
fact that if they do not measure the very closest known pairs
these must go unmeasured.*
* Some of them may, however, be measured with the interferometer.
The method is described on p. 67.
OBSERVING METHODS, VISUAL BINARY STARS 59
EYEPIECES
The power of the eyepiece to be used is a matter of practical
importance, but one for which it is not easy to lay down spe
cific rules. The general principle is — use the highest power the
seeing will permit. When the seeing is poor, the images "danc
ing" or blurred, increase in the magnifying power increases
these defects in the images and frequently more than offsets
in this way the gain from increase in the scale. On such nights,
if they are suitable for any work, choose wider pairs and use
lower powers. The practical observer soon realizes that it is
not worth while to measure close pairs except with high powers.
With the 36in. telescope my own practice is to use an eyepiece
magnifying about 520 diameters for pairs with angular separation
of 2" or more. If the distance is only 1", I prefer a power of
1,000, and for pairs under 0"5 I use powers from 1,000 to 3,000,
according to the angular distance and the conditions. The
closeness and brightness of the pair and the quality of the
definition are the factors that determine the choice. Very
close pairs are never attempted unless powers of 1,500 or higher
can be used to advantage.
The simplest method of measuring the magnifying power of
an eyepiece in conjunction with a given objective is to find
the ratio of the diameter of the objective to that of its image
formed by the eyepiece — the telescope being focused and directed
to the bright daylight sky. Two fine lines ruled on a piece of
oiled paper to open at a small angle form a convenient gage
for measuring the diameter of the image. A very small error
in this measure, however, produces a large error in the ratio
and the measure should be repeated many times and the mean
result adopted. The magnifying power of an eyepiece may,
of course, also be measured by a dynameter if one is available.
DIAPHRAGMS
It is sometimes said that the quality of star images is improved
by placing a diaphragm over the objective to cut down its
aperture. I question this. It is certain that the experience
of such observers as Schiaparelli and Burnham was directly
opposed to it, and experiments made with the 12in. and 36in.
telescopes offer no support for it. Indeed, it is difficult to
understand how cutting off part of the beam of light falling
60 THE BINARY STARS
upon an object glass of good figure can improve the character
of the image, unless it is assumed that the amplitude of such
atmospheric disturbances as affect the definition is small enough
to enter the problem. The only possible gain might be in the
reduction of the brightness of the image when one star of a pair
is exceptionally bright, as in Sirius; but this reduction can be
effected more conveniently by the use of colored shade glasses
over the eyepiece. These are occasionally of advantage. *
A hexagonal diaphragm placed over the objective, however,
may prove of great value in measuring stars, like Sirius or
Procyon, which are attended by companions relatively very
faint; but this is because such a diaphragm entirely changes
the pattern of the diffraction image of the star, not because
it cuts down the aperture of the telescope. The pattern is
now a central disk from which six thin rays run; between these
rays the field appears dark even close to the bright star, and
a faint object there can be seen readily that would be invisible
otherwise. Professor Barnard f used such a diaphragm to
advantage with the 40in. Yerkes refractor. Provision should
be made for rotating the diaphragm through an angle of about
60° and it will be convenient in the case of a large instrument
to be able to do this by means of gearing attached to a rod
running down to the eye end.
ERRORS OF OBSERVATION
All measures of angles or of distances are affected by errors,
both accidental and systematic, and when, as in double star
work, the measured quantities are very minute, these errors
must be most carefully considered. The accidental errors
may be reduced by careful work and by repeating the meas
ures a suitable number of times. Little is to be gained, in this
respect or in any other, by making too large a number of set
tings upon an object on any one night; because such factors
as the seeing, the hour angle, the observer's physiological con
dition, all remain nearly constant. As a rule, four settings for
position angle and three measures of double distance are enough
* Van den Bos, in a recent letter, argues that under poor atmospheric
conditions, an iris diaphragm, to reduce aperture, is helpful, particularly
in the measurement of unequal or very bright pairs.
t A.N. 182, 13, 1909.
OBSERVING METHODS, VISUAL BINARY STARS 61
to make on one night, but the measures should be repeated
on one or more additional nights. This is not only to reduce
the accidental error of measure but to guard against outright
mistakes in reading the circles, recording, etc. As to the number
of nights on which a system should be measured at a given epoch,
opinions will differ. Some observers run to excess in this matter.
Generally, it may be said that it is time wasted to measure a
system on more than four nights at any epoch and ordinarily the
mean of three nights' measures, or even of two, if the pair is easy
to measure and the measures themselves are accordant, is as
satisfactory as the mean from a larger number. In critical cases,
however, a larger number is sometimes desirable.
The systematic errors of measurement are far more trouble
some, for they vary not only with the individual but are differ
ent for the same observer at different times and for different
objects. Aside from the personality of the observer, they
depend upon the relative magnitudes of the two components
of a double star, the angular distance, the angle which the line
joining the stars makes with the horizontal, and, in unequal
pairs, upon the position of the faint star with respect to the
bright one. Various methods have been adopted to deter
mine these errors or to eliminate them.
The most elaborate investigation in this line is probably
the one made by Otto Struve, who measured "artificial double
stars formed by small ivory cylinders placed in holes in a black
disk." He deduced formulae by means of which he calculated
corrections to be applied to all his measures; but it is very
doubtful whether these corrections really improve the results.
I agree with Lewis when he says, "I would prefer his original
measures— in part because the stars were so particularly arti
ficial." The actual conditions when observing the stars at
night are of necessity widely different from those under which
the test measures were made. Certainly, in the case of Otto
Struve, the corrected angles and distances are frequently more
at variance with the general run of all of the measures by good
observers than the original values. The student of double
star motions will generally find it advantageous to use the
original uncorrected measures of every observer in his pre
liminary work and then to derive values for the systematic
or personal errors of each by comparing his measures with
the curve representing the means of all available measures.
62
THE BINARY STARS
The observer, on the other hand, may profitably adopt
observing methods designed to eliminate, in part at least,
systematic errors. Innes' plan of measuring each pair on
each side of the meridian is an excellent one because, in gen
eral, the line joining the two stars changes its angle with respect
to the horizon in passing the meridian. In the extreme case,
if the smaller star is above the primary when the pair is east
of the meridian, it will be below when west of the meridian.
When Innes' two measures made in this way were not sufficiently
accordant, he repeated them on two additional nights, one night
in each position of the instrument.
In 1908, MM. Salet and Bosler* published the results of an
investigation of the systematic errors in measures of position
angle in which they made use of a small total reflecting prism
mounted between the eyepiece and the observer's eye and
capable of being rotated in such manner as to invert the field
of view. Theoretically, half the sum of the measures made
without and with the prism should represent the angle freed
from errors depending upon the inclination of the images to
the horizon. In fact, Salet and Bosler found that, whereas
their measures without the prism and those made with it both
showed a personal equation varying in amount with the star,
the means of the two sets were remarkably free from person
ality. Here, for example, are their measures of y Leonis:
Observer
Date
Without prism
With prism
Mean
Salet
Bosler
(SB)
1907.19
1907.23
11904
116.80
+2.24
11350
116.07
2.57
11627
116.44
0.17
Hermann Struve and J. Voute later published measures made
in this manner and each concluded that the results were far better
than his measures made entirely without the use of the prism.
In one of his papers t Voute states that "it is principally in
observing in the perpendicular ( :) position that the observations
show a pronounced systematic error," while "the parallel (. .)
observations are in general free from systematic errors."
* Bull. Astronomique 26, 18, 1908.
t Union Obs. Circ. 27, 1915.
OBSERVING METHODS, VISUAL BINARY STARS
63
Dawes* long ago pointed out that in "rather close double
stars," the measures of distance "will almost inevitably be
considerably too large" unless the observer has taken into
account the change made in the apparent form of the star
disk when a thread of the micrometer is placed over it. This
change is in the nature of a swelling out of the disk on each
side of the thread, producing an approximately elliptical disk.
When two images are nearly in contact and the threads are
placed over them, this swelling obliterates the interval between
the disks and the threads are therefore set too far apart. The
effect disappears when the disks are well separated.
In my investigations of double star orbits I have frequently
noticed that distance measures of a given system made with
small apertures are greater than those made with large telescopes
even when made by the same observer, provided the system is a
close one as viewed in the smaller instrument. I have found such
a systematic difference in the distances in stars which I have
measured with the 12in. and with the 36in. telescope, and
Schlesingerf has also called attention to this difference, giving a
table derived from my measures as printed in Vol. XII of the
Publications of the Lick Observatory. This table is here reproduced
with a column of differences added :
Measured Separations
Number of stars
With the 12in.
With the 36in.
Difference
20
0''52
0'42
H0'.'IO
25
0.62
0.54
+0.08
20
0.71
0.64
+0.07
24
0.81
0.79
+0.02
24
1.07
1.03
+0.04
21
1.38
1.39
0.01
26
2.13
2.10
+0.03
18
4.49
4.53
0.04
The systematic difference is clearly shown in all the pairs
having a separation less than twice the resolving power (0"42)
of the 12in. telescope; in the wider pairs it is negligibly small.
Occasionally an observer's work shows systematic differences
of precisely the opposite sign. Thus Schlesinger (loc. cit.)
* Mem. R. A. S., 35, 153, 1867.
t Science, N. S., 44, 573, 1916.
64 THE BINARY STARS
in analyzing the measures by Fox in the Annals of the Dearborn
Observatory, Vol. I, (1915) finds that the distances are measured
smaller with the 12in. than with the 182^in. or with the 40in.,
"the differences being largest for small separations and becoming
negligibly small for separations in the neighborhood of 5"."
The personal equation revealed in such comparisons as these
must obviously be taken into account in orbit computations.
PHOTOGRAPHIC MEASURES
The first double star to be discovered visually was also the
first for which measurable photographic images were obtained.
G. P. Bond photographed f Ursae Majoris, angular separation
14'.'2, on a collodion plate in 1857, giving an 8sec. exposure.
Pickering and Gould in America, M. Henry in France, and the
Greenwich observers in England, among others, followed up
this early success and secured results of value for a number of
pairs, a few of them as close as 1". More recently, extensive
programs have been carried out at several observatories and it
has become evident that photographic measures, made under
proper conditions, are of the same order of accuracy as visual
ones for pairs with angular distances of from 1" to 2", if the
components are not very unequal in magnitude, and that they
exceed visual measures in accuracy for wider pairs. A photo
graphic observation of a double star, moreover^ has the great
advantage, as Hertzsprung has pointed out, of being a "per
manent document " which can be reexamined as often as may be
desirable.
Hertzsprung, working at Potsdam in the years 1914 to 1919,
made a thorough investigation of the possibilities of the photo
graphic method as applied to the measurement of double
stars, including a study of the sources of accidental and system
atic errors, and of the procedure necessary to eliminate such
errors or, failing this, to reduce to a minimum their effects upon
the measures.
It appears that the chief sources of error are (1) the difference
in magnitude of the components of a pair, (2) refraction, and
more particularly, the change of refraction with the color of a
star, and (3) the Eberhard effect, or the effect upon a photo
graphic star image produced by the close proximity of another
star image. There are, of course, also the accidental errors of
measurement, and the errors arising from incorrect scale value
OBSERVING METHODS, VISUAL BINARY STARS 65
and imperfect orientation, errors that are comparable to those
in visual observations from corresponding sources.
1. The Magnitude Equation. — The images of the two com
ponents of a double star are so nearly in contact on the photo
graphic plate that the methods of eliminating the error arising
from difference in magnitude in use in parallax determinations
and other precise photographic measurements are not applicable.
A rotating sector, for example, would, for all but the relatively
wide pairs, cut down the light from both components in sub
stantially the same degree.
The method that has given the best results up to the present
time is the one adopted by Hertzsprung and others of covering
the objective of the telescope with a coarse grating made of rods
of uniform thickness, uniformly spaced. Such a grating will
give, in addition to the principal image of a star, spectra sym
metrically placed on either side of it, and if it is made so coarse
that the firstorder spectra are just clearly separated from the
principal image, they, too, will have sensibly stellar images.
Further, by proper choice of rod thickness and free spacing the
difference in magnitude between the principal image and its
firstorder spectra may be made to vary from one to four or more
magnitudes. It is well to have a set of six or more such gratings
so calculated that the magnitude difference may be varied by
steps of approximately a magnitude. It will then be possible
in practically all cases to bring the brightness of the firstorder
spectra of the primary within half a magnitude of that of the
principal image of the companion. A difference of half a
magnitude will have little or no effect upon the measures. The
gratings, it is to be noted, will reduce the brightness of the
principal images of both components by an amount depending
upon the grating constant, and the exposure time will also vary
with the grating used.
The practical difficulties in using this method with telescopes of
large aperture arise from the expense involved in constructing
good gratings of large size and from the inconvenience of inter
changing them in the course of a night's observing.
2. Refraction and Star Color. — On account of atmospheric
dispersion, the refraction varies with the effective wave length
of a star's light. A systematic error of measurement may,
therefore, result when the colors of the two components of a
double star differ appreciably. To eliminate it, take the photo
66 THE BINARY STARS
graphs in light of a restricted range in wave length, by using
appropriate niters and, if necessary, special plates. The particu
lar niters and plates required will vary with the telescope used.
It is desirable also, to avoid taking photographs at great zenith
distances.
3. The Photographic Star Image. — Eberhard, many years
ago, showed that the work done by a developer in blackening
any small area on a photographic plate depended upon how
much more work it had to do in the immediate vicinity. The
density of a star image, for a given exposure and development,
therefore, depends not only upon the brightness of the star but in
part also upon the presence of other star images in its immediate
proximity. In the case of a double star the outer edge of the
image of each component will be denser than its inner edge —
the edge nearest the other component. The effective centers
of the two images are therefore displaced in opposite directions
and the apparent or measured distance is larger than the true
distance between them. But Hertzsprung also found evidence
of a displacement in the opposite sense in the case of some photo
graphs of very close pairs taken under relatively poor conditions.
Apparently, however, both effects appear only in pairs in which
the images of the two components are nearly or quite in contact.
For pairs with clearly separated images, the error from this source
is so small that it may be neglected.
In addition to the precautions to be taken with respect to the
sources of systematic error just described, it is, of course, neces
sary to put the plate accurately in focus, and to determine the
parallel carefully. To provide means for determining the
parallel a bright star in the immediate neighborhood may be
allowed to trail on the plate, or images of the double star (with
the plate displaced slightly in declination) may be taken both
near the preceding and near the following edge of the plate,
the clock being stopped between the two exposures. The
exposure time will depend upon the star and the telescope.
It is desirable to have distinct images but not denser than
necessary for easy measurement.
Finally, the importance of photographing only on calm nights,
and of having an accurate driving mechanism for the telescope
is to be particularly emphasized.
The plates thus secured should be measured in rectangular
coordinates and it will be advantageous to use an engine in which
OBSERVING METHODS, VISUAL BINARY STARS 67
the plate, rather than the eyepiece, is moved by the micrometer
screw. In this case the value of a revolution of the screw, in
seconds of arc, is independent of the power of the eyepiece, and
the plate may be measured both film side up and through the
glass. It is hardly necessary to add that the scale value of the
plates must be determined accurately and that the plates must be
oriented with great care. Since the firstorder spectra are
symmetrically placed with respect to the center of the principal
image, the mean of their positions may be adopted as the position
of the primary.
It has already been noted that in recent years lists of new
double stars detected on parallax plates and lists of others found
in the examination of the astrographic star catalogue for different
zones have been published by Scheiner, Stein, Barton, Olivier,
and others. The pairs in the astrographic lists are all relatively
wide and for the most part faint, and the measures of position
angles and distances are not of a high order of accuracy. The
lists from parallax plates also consist of faint pairs, but some of
them have angular distance not greatly in excess of a second
of arc. The measures of these pairs are probably quite as
accurate as visual measures would be.
INTERFEROMETER MEASURES
The advantages of the photographic method are so great that
it will undoubtedly come into ever more general use in the
measurement of all double stars that can be resolved on the
photographic plate. The closer pairs, and particularly those
whose components differ greatly in magnitude, must be left to
the visual observer working with the filar micrometer, with or
without the reversing prisms, or instruments that are improve
ments upon it. Hargreaves, for example, has recently designed
a comparison image micrometer* that promises to give more
accurate results, for the angular distance, at least, than the
filar micrometer.
The lower limit for accurate micrometric measurements of
angular separation with existing telescopes is about 0''13,
though under the best conditions fairly good estimates may be
made for pairs as close as O''IO. The only instrument that
promises good results for still closer pairs is the interferometer.
* See footnote, p. 41.
68 THE BINARY STARS
Michelson, more than 40 years ago, showed that such extremely
small angles as the diameter of a small satellite or the distance
between the components of a double star could be measured
"by observing the interference fringes produced at the focus
of a telescope when only two portions of the objective, located
on the same diameter, are used." He demonstrated that "as
the distance apart of the apertures is increased the visibility of
the fringes reaches a minimum for a distance equal to 1.22X/a,
or 0.5X/a, for a disk or double star, respectively, where X is the
effective wave length and a is the desired angle." He applied
the method to the measurement of the diameters of the satellites
of Jupiter, with the 12in. refractor of the Lick Observatory
in 1891. Four years later,* Schwarzschild applied the principle
to the measurement of double stars, using a specially designed
interferometer placed over the objective of the 10in. refractor
at Munich. He measured a number of pairs with angular
distances ranging from 0"9 to 3"7. No one else seems to have
made interferometer measures of double stars until Anderson,f
early in 1920, using an interferometer of his own design, attached
to the 100in. reflector at Mount Wilson, measured the position
angle and distance of Capella, with an accuracy far exceeding
the accuracy of the best micrometer measures of a close double
star. Merrill,! using the same instrument, secured additional
measures of Capella later in the same year and in 1921, and also
measured k Ursae Majoris = A1585, which he and Anderson
had discovered as a double, by the use of the interferometer, before
they knew of my earlier micrometric measures. My measures
had shown a rapid decrease in angular distance, the value in
1919.29 being 0''15. Their measures, in 1921, gave 0''08,
with a change in the position angle that was consistent with the
law of areas. Attempts to find other double stars by examination
with the interferometer gave negative or doubtful results, and
the work had presently to be discontinued.
Experiments made at the Lick and at the Yerkes Observatories
did not prove satisfactory. At Catania, M. Maggini,§ in 1922,
applied an interferometer to the 12in. equatorial and in the
years immediately following made a considerable number of
* A.N. 139, 353, 1896.
t Contr. ML Wilson Obs. 9, 225, 1920.
t Contr. ML Wilson Obs. 11, 203, 1922.
§ Pub. R. Oss. Catania, 1925.
OBSERVING METHODS, VISUAL BINARY STARS 69
measures of double stars with angular distances up to 0"25 and
even greater. These pairs were all within the measuring range
of the micrometer attached to larger telescopes, and a com
parison of actual measures could thus be made. In the majority
of cases, Maggini's results were in fair to good accord with those
made with the micrometer, but in a number of cases, the dis
agreement was far beyond the error of micrometric measurement,
and his values were entirely out of accord with the earlier (and
later) micrometric results, though the pairs in question were
as well suited to measurement by both methods as the others
on his list. It is clear that further investigation is required
before the interferometer can be accepted as a standard instru
ment for double star observation. We must know how to
distinguish between genuine fringes and fringe disappearance
and spurious effects of any kind whatever; and we must determine
the limits of magnitude and angular separation within which the
interferometer can be applied successfully with a telescope of
given aperture and focal length. It may be found that the
number of pairs that can be measured with the interferometer
is small, but, by way of compensation, these will, in general,
be pairs that cannot be measured successfully by any other
method unless the inclination of the orbit plane is high enough
to permit measures of the relative radial velocities of the two
components. It is greatly to be desired that experiments with
the interferometer be continued.
[The reader is referred to the second volume of the series Stars and Stellar
Systems edited by Gerard P. Kuiper and Barbara M. Middlehurst. Volume
II, Astronomical Techniques, was edited by William A. Hiltner. Chapter 19,
"Techniques for Visual Measurements," by P. Muller, is a very fine discussion
of instruments and their uses as related to visual binary stars. Chapter 22,
by W. H. van den Bos, "Orbit Determinations of Visual Binaries"; chapter
23, "Spectroscopic Binaries," by R. M. Petrie; and chapter 24, "Eclipsing
Binaries," by John B. Irwin, are excellent treatments of their subjects.
Volume III of this same series, to be published sopn, will also contain much
relevant material. — J.T.K.]
CHAPTER IV
THE ORBIT OF A VISUAL BINARY STAR
We have seen that Sir William Herschel, by his analysis of
the observed motion in Castor and other double stars, demon
strated that these systems are "real binary combinations of
two stars, intimately held together by the bonds of mutual
attraction." Later observation has shown that the apparent
motion in such systems is on the arc of an ellipse and that the
radius vector drawn from the primary star to its companion
sweeps over areas which are proportional to the times. It has
therefore been assumed from the beginning that the attractive
force in the binary star systems is identical with the force of
gravitation in our solar system, as expressed by Newton's law,
and the orbit theories which we are to investigate in the present
chapter are all based upon this assumption. Before taking
up the discussion of these theories it is pertinent to inquire
whether the fundamental assumption is justified.
It is supported by all of the available evidence, but rigorous
mathematical proof of its validity is difficult because the
motion which we observe in a stellar system is not the true
motion but its projection upon a plane perpendicular to the
line of sight. The apparent orbit is therefore, in general, not
identical with the true orbit and the principal star may lie at
any point within the ellipse described by the companion and
not necessarily at either the focus or the center. Hence, in
Leuschner's words, "mathematical difficulties are encountered
in establishing a law of force which is independent of the angle
0, the orientation." In the article quoted, Leuschner, after
pointing out that "Newton did not prove the universality of
the law of gravitation, but by a happy stroke of genius gener
alized a fact which he had found to be true in the case of the
mutual attraction of the Moon and the Earth," proceeds to
show that the law does hold throughout the solar system, the
question of orientation not entering. He then says that, in
binary systems, "when the law is arbitrarily assumed to be
70
THE ORBIT OF A VISUAL BINARY STAR 71
independent of the orientation, as was found to be the case in
the solar system, two possibilities arise, namely, either that
the force is in direct proportion to the distance r between the
two stars or that the Newtonian law applies. It can be shown,
however, that when, in the case of an elliptic orbit, the force
is proportional to r, the primary star must be in the center of
the ellipse. As this has never been found to be the case, the
only alternative is the Newtonian law."
It should be clearly understood that the difficulty in demon
strating the universality of the law of gravitation here pointed
out is purely mathematical. No physical reason has ever been
advanced for a dependence of an attracting central force upon
the orientation, and until such dependence has been proved
we may safely proceed with our investigation of binary star
orbits under the action of the law of gravitation.
Until the relative masses of the two components are known
it is impossible to determine the position of the center of gravity
of the system and we are therefore unable to compute the
orbits described by the two stars about that center. What
our measures give us is the apparent orbit of one star, the
companion, described about the other, the primary, which is
assumed to remain stationary at the focus. It is clear that
this relative orbit differs from the actual orbits of the two com
ponents only in its scale.
The problem of deriving such an orbit from the micrometer
measures of position angle and distance was first solved by
Savary,* in 1827, but Encke f quickly followed with a different
method of solution which was somewhat better adapted to
the needs of the practical astronomer, and Sir John HerschelJ
communicated a third method to the Royal Astronomical
Society in 1832. Since then the contributions to the subject
have been many. Some consist of entirely new methods of
attack, others of modifications of those already proposed.
Among the more notable investigators are Villarceau, Madler,
Klinkerfues, Thiele, Kowalsky, Glasenapp, Seeliger, Zwiers,
Howard, Schwarzschild, See, Russell, Innes, and van den Bos.
The methods of Savary and Encke utilize four complete
measures of angle and distance and, theoretically, are excellent
* Savary, Conn, des Temps, 1830.
t Encke, Berlin Jahrbuch, 1832.
 Herschel, Memoirs R.A.S., 6, 171, 1833.
72 THE BINARY STARS
solutions of the problem; Herschel's method is designed to utilize
all the available data, so far as he considered them reliable.
This idea has commended itself to all later investigators.
Herschel was, convinced, however, that the measures of dis
tance were far less trustworthy than those of position angle,
and his method therefore uses the measures of distance simply
to define the semimajor axis of the orbit; all of the other ele
ments depend upOn measures of position angle. At the time
this may have been the wisest course, but the distance meas
ures of such early observers as W. Struve, Dawes, and Dem
bowski, and those of later observers working with modern
micrometers, are entitled to nearly or quite as much weight as
the measures of position angle and should be utilized in the
entire orbit computation.
Whatever method is adopted, the investigator must, of course,
begin by assuring himself that he has data sufficient for a satis
factory computation. In deciding this, he should consider both
the length of the observed arc and its form. With strongly
marked curvature, a comparatively short arc may suffice,
provided s the observations have a high degree of accuracy.
Ordinarily, however, the arc should be long enough to cover
both ends or elongation points of the apparent ellipse.
Satisfactory data being given, the problem before the com
puter evidently consists of two parts: first, the determination of
the apparent ellipse, or the constant of areal velocity, from the
data of observation ; secondly, the derivation of the elements
of the true orbit from the relations between an ellipse and its
orthographic projection.
THE APPARENT ELLIPSE
Every complete observation of a double star supplies us with,
three data: the time of observation, the position angle of the
companion with respect to the primary, and the angular dis
tance between the two stars. It is clear, as Comstock pointed
out many years ago, that the time of observation is known
with far greater accuracy than either of the two coordinates
of position. The relations between the times of observation
and the motion in the ellipse should therefore be utilized; that
is, the condition should be imposed that the law of areal veloc
ities must be satisfied as well as the condition that the points of
observation should fall approximately upon the curve of an
THE ORBIT OF A VISUAL BINARY STAR 73
ellipse. Elementary as this direction is, it is one that has been
neglected in many a computation.
Theoretically, the first step in our computation should be the
reduction of the measured coordinates to a common epoch by
the application to the position angles of corrections for pre
cession and for the proper motion of the system. The distance
measures need no corrections. Practically, both corrections
are negligibly small unless the star is near the Pole, its proper
motion unusually large, and the time covered by the observa
tions long. The precession correction, when required, can be
found with sufficient accuracy from the approximate formula
A0 = +0?0056 sin a sec 8 (t  t ) (1)
The formula for the correction due to the proper motion of
the system is
Ad = //' sin 8 (t  to) (2)
where /z" is the proper motion in right ascension expressed in
seconds of arc*
When the measures of any binary star have been tabulated
(with the above corrections, if required), they will exhibit
discordances arising from the accidental and systematic errors of
observation and, occasionally, from actual mistakes. If they are
plotted, the points will not fall upon an ellipse but will be joined
by a very irregular broken line indicating an ellipse only in
a general way. It will be advisable to investigate the meas
ures for discordances before using them in the construction
of the apparent ellipse and the simplest method is to plot
upon coordinate paper first the position angles and then the
distances, separately, as ordinates, against the times of obser
vation as abscissae, using a fairly large scale. Welldetermined
points (for example, a point resting upon several accordant
measures by a skilled observer and supported by the preceding
and following observations) may be indicated by heavier marks.
Smooth freehand curves, interpolating curves, are now to be
drawn to represent the general run of the measures and in
drawing these curves more consideration will naturally be
given to the well observed points than to the others. Obser
* See "Note on the Effect of Proper Motion on Double Star Measures,"
by Alan Fletcher (Mon. Not. R.A.S. 92, 119, 1931), for a more complete
discussion.
74 THE BINARY STARS
vations that are seriously in error will be clearly revealed and
these should be rejected if no means of correcting them is avail
able. The curves will also show whether or not the measures
as a whole are sufficiently good to make orbit computation
desirable.
If the amount of available material warrants it, the question
of the systematic or personal errors of the observers should
also be considered at this time. No reliable determination of
such errors is possible unless (a) measures by the same observer
under essentially the same conditions in at least four or five
different years are at hand, and (6) unless the total number of
measures by many different observers is sufficient to establish
the general character of the curves beyond reasonable question.
If the second condition is satisfied, the average of the residuals
from the curve for a given observer may be regarded as his
personal error and the corresponding correction may be applied
to all of his measures. Two further points should be noted:
First, the residuals in position angle should be reduced to arc
by multiplying by the factor p/57.3 before the mean is taken,
to allow for the effect of variations in the angular separation;
second, the corrections should not be considered as constant
over too long a period of time. The application of such correc
tions has the effect of reducing the size of the residuals, but the
principal advantage to be gained from it is that it lessens the
danger of giving undue weight to measures seriously affected by
systematic errors of observation.
After all corrections have been applied, the measures which
are retained should be combined into annual means or into
mean places at longer or shorter time intervals according to
the requirements of the particular case. Several factors really
enter into the question of the weights to be assigned to the
individual observations in forming these means; for instance,
the size of the telescope used, the observing conditions, the
number of nights of observation, and the experience of the
observer; but it will be wise, in general, to disregard all but
the number of nights of observation, provided the telescope
used is of adequate resolving power for the system in question
and that the observer has not specifically noted some of his
measures as uncertain. A single night's measure deserves
small weight; mean results based upon from two to six nights'
accordant measures may be regarded as of equal weight; means
THE ORBIT OF A VISUAL BINARY STAR 75
depending upon a much larger number of measures may be
weighted higher. In general, a range in weights from one to
three will be sufficient.
If, in addition to the visual measures, photographic measures
have been made at several epochs, the combination of the
measures by the two different methods merits special considera
tion. When, as, for example, in the case of $ Ursae Majoris,
many photographic measures of great accuracy are available,
these should unquestionably be given high weight. If, on the
other hand, only a few photographic measures have been made
they may be weighted on the same basis as the visual observations.
Having thus formed a series of normal places, we may find
the apparent ellipse that best represents them either graphi
cally or by calculating the constants of the general equation
of the ellipse with the origin at any point. This equation is
ax 2 + 2hxy + by 2 + 2gx + 2fy + c = (3)
which may be written in the form
Ax 2 + 2Hxy + By 2 + 2Gx + 2Fy + 1 = (4)
in which we must have A > 0, B > 0, and AB  H 2 > 0.
If we assume the position of the primary star as origin, we
may calculate the five constants of this equation from five
normal places by the relations
x = p sin d) ^
y = p cos 8j
but it is advisable to make a least squares solution using all of
the normal places.
The great objection to this method is that it entirely disre
gards the time of observation. Moreover, the errors of obser
vation, small as they are numerically, are large in proportion
to the quantities to be measured, a fact that makes it difficult
to obtain a satisfactory ellipse without repeated trials. The
graphical methods are therefore to be preferred.
The simplest method, and one that in most cases is satis
factory, is to plot the positions of the companion star in polar
coordinates, the primary star being taken as the origin. With
the aid of an ellipsograph or by the use of two pins and a thread,
an ellipse is drawn through the plotted points and is adjusted
by trial until it satisfies the law of areas. This adjustment
76 THE BINARY STARS
must be made with the greatest precision and the curve of the ellipse
drawn with great care, for the construction of the apparent
ellipse is the critical part of the entire orbit determination.
In my own practice I have found that the test for the law of
areas can be made most rapidly by drawing radii to selected
points which cover the entire observed arc and measuring
the corresponding elliptic sectors with a planimeter. The
comparison of the areal velocities derived from the different
sectors at once indicates what corrections the ellipse requires.
With a suitable ellipsograph a new ellipse is quickly drawn
and the areas again measured. The process is repeated until
a satisfactory ellipse has been obtained.
Some investigators prefer the mode of procedure in con
structing the apparent ellipse first suggested by Sir John Herschel.
An interpolating curve is drawn, in the manner described
above, for the position angles only, using the mean or normal
places. If the curve is carefully drawn, smoothly and without
abrupt changes of curvature, it should give the position angle for
any particular epoch more accurately than the measure at
that epoch, for it rests upon all of the measures. From this
curve read the times corresponding to, say, every 5° of angle,
tabulate them, and take the first differences. Dividing these
by the common angle difference will give a series of approximate
dd
values of dt/dd. But by the theory of elliptic motion pSr
must be a constant and hence p = c^/^ Therefore a series
\ ad
of relative values of the distance (expressed in any convenient
unit) corresponding to every fifth degree of position angle
can be derived from the table of angles. Now plot the points
representing corresponding angles and relative distances; if
the interpolating curve has been correctly drawn and read
off they will all lie upon the arc of an ellipse. If they do not,
draw the best possible ellipse among them and use it to correct
the interpolating curve, repeating the process until the result
is satisfactory. Finally, convert the relative into true distances
by comparing those distance measures which are regarded as
most reliable with the corresponding values in the unit adopted
in the plot.
There are at least two objections to this method: First, it
does not make adequate use of the observed distances; and
THE ORBIT OF A VISUAL BINARY STAR 77
second, when the angle changes rapidly, as it does in many
systems at the time of minimum apparent separation, it is
almost impossible to draw the interpolating curve correctly.
The former difficulty may in many cases be overcome by the
following procedure: Read from the interpolating curve for
position angles, normal values for every fourth year.* Regard
each two consecutive values as the limiting radii of a circular
sector, and as the radius of each sector adopt its mean observed
distance, which may be derived, if desired, from an interpolating
curve. Draw these circular sectors on coordinate paper and
pass through them a freehand curve, approximately the arc of
an ellipse, giving sectors of equal area. In general, however,
it is in my judgment most satisfactory to plot the positions of the
companion star directly in polar coordinates, using normal
places. From these the ellipse that best satisfies the law of
areas must be found by the method of trial and correction.
THE TRUE ORBIT
After the apparent ellipse has been constructed graphically,
or from the constants in the equation of the ellipse, it remains
to derive the elements which define the form and size of the
true orbit, the position of the orbit plane, the position of the
orbit within that plane, and the position of the companion
star in the orbit at any specified time. Three of the required
elements have dynamic significance and are entirely independent
of the space location of the system. These are the revolution
period, the time of periastron passage, and the eccentricity.
To these belongs the semimajor axis of the orbit when measured
in linear units; measured in angular units it is, of course, also a
function of the parallax of the system. The three remaining
elements, the inclination of the orbit plane, the position of the
line of nodes, and the angle between that line and the major
axis are purely geometric and merely relate the orbit of the double
star system to the orbit of the Earth.
The first four elements may be defined formally as follows : Let
P = the period of revolution expressed in mean solar years.
T = the time of periastron passage.
e = the eccentricity.
a = the semiaxis major expressed in seconds of arc.
* This will suffice for longperiod systems; for systems of short period,
readings at shorter intervals should be taken.
78 THE BINARY STARS
All authorities are agreed upon these definitions, but some
confusion in the nomenclature and even in the systems used in
defining the remaining elements has arisen from the fact that
it is impossible to say, from the micrometer measures alone,
on which side of the plane of projection (which is taken as the
plane of reference) the companion star lies at a given time.
In other words, we cannot distinguish between the ascending
and the descending node, or between direct and retrograde
motion in the ordinary sense. Further, in some systems the
observed position angles increase with the times, in others they
decrease.
The following is the classical system in the form most con
venient when the requirements of the observer of radial velocities
are considered as well as those of the observer with the microm
eter. Let
12 = the position angle of that nodal point which lies between
0° and 180°; that is, the position angle of the line of
intersection of the orbit plane with the plane perpendicular
to the line of sight. Call this merely "the nodal point,"
disregarding the distinction between ascending and
descending nodes,
w = the angle in the plane of the true orbit between the line
of nodes and the major axis. It is to be measured from
the nodal point to the point of periastron passage in
the direction of the companion's motion and may have
any value from 0° to 360°. It should be stated whether
the position angles increase or decrease with the times.
i = the inclination of the orbit plane; that is, the angle
between the orbit plane and the plane at right angles to
the line of sight. Its value lies between 0° and ±90°
and may carry the double sign ( + ) or be left without
sign until the indetermination has been removed by
measures of the radial velocity.* When these are
available, i is to be regarded as positive (+) if the orbital
motion at the nodal point is carrying the companion
star away from the observer; negative, if it is carrying
the companion star toward the observer.
* Van den Bos, however, writes i in the second quadrant in all orbits in
which the position angle decreases with the time ("A Table of Orbits of
Visual Binary Stars," B.A.N. 3, 149, 1926), and some later computers have
followed this convention.
THE ORBIT OF A VISUAL BINARY STAR 79
The symbol » denotes the mean annual motion of the
companion, expressed in degrees and decimals, measured
always in the direction of motion.
The conventions of taking ft always less than 180° and of
counting « (for which many computers use the symbol X) always
in the direction of the companion's motion were first suggested,
I believe, by See, and have now been adopted generally. The
definition of i (for which some computers write 7) is the usual
one, also. Many computers prefer to count the mean annual
motion in the direction of increasing position angles in all systems,
and to consider the motion negative when the angles decrease
with the times.
As early as 1883 T. N. Thiele* proposed a method of computa
tion that would replace the geometrical elements i, a>, and ft
and the semiaxis major a by the polar coordinates, with respect
to the center of the apparent ellipse, of the projections on the
plane of the apparent orbit of the two points in the true orbit
for which the eccentric anomalies are, respectively, 0° and 90°.
More recently Innes independently worked out a system which
is, in all essentials, the same as Thiele's. Innes' system, as
formulated by van den Bos, is given on a later page and the
relations between the ThieleInnes constants and the elements
as defined in the preceding paragraphs are there set forth.
When the elements are known, the apparent position angle
and the angular distance p for the time t are derived from
the following equations:
360°
M = ti(t  T)  E  e sin E(
r = a(l — e cos E)
tan Y 2 v = J jjr^ tan Y 2 E
tan (0 — ft) = ± tan (v + w) cos i
p = r cos (v + «) sec (0 — ft)
(6)
} (7)
Equations (6) are the usual ones for elliptic motion, the
symbols M, E, and v representing respectively, the mean,
eccentric, and true anomaly, and r the radius vector. Equa
tions (7) convert the v and r of the companion in the true orbit
* A.N. 104, 245, 1883.
80 THE BINARY STARS
into its position angle and distance in the projected, or apparent
orbit. Position angles are generally recorded only to the
nearest tenth of a degree in orbit computation, hence it is
sufficiently exact to take the value of E corresponding to a
given value of M from. Astrand's Hiilfstafeln, which hold for
all values of the eccentricity, or the value of v directly from
the still more convenient Allegheny Tables,* provided the eccen
tricity does not exceed 0.77. If the latter tables are used,
it is convenient to derive the value of r from the equation
r _ «(ls') m
(1 + e cos v)
instead of from the third of Eqs. (6).
KOWALSKY'S METHOD
From the many methods of orbit computation that have been
formulated, I have selected for presentation here those by
Kowalsky, by Zwiers and by Innes. All three are of very general
application and each one has its advocates among computers.
Several other methods are useful and the student is advised to
examine all those for which references are given at the end of
this chapter f
Kowalsky's methodj is essentially analytical and derives the
orbit elements from the constants of the general equation of
the apparent ellipse which is the orthogonal projection of the
true orbit, the origin of coordinates being taken at S, the position
of the primary star. This equation takes the form
Ax 2 + 2Hxy + By 2 + 2Gx + 2Fy + 1 =
The values of the constants A, H, B, G and F, may be com
puted by the method noted on page 75, but this is open to the
serious objection that it takes no account of the law of areas.
It is far better to follow the procedure proposed by Glasenapp§
and derive the values of the five constants from measures on the
carefully drawn apparent ellipse, as follows :
* Pvbl. Allegheny Obs. 2, 155190, 1912.
t See particularly the methods of Russell (1933) and of Volet (1932).
% First published, according to von Glasenapp, in the Proceedings of the
Kasan Imperial University, in 1873. This volume has not been accessible
to me.
§ Mon. Not. R.A.S. 49, 276, 1889.
THE ORBIT OF A VISUAL BINARY STAR 81
In the general equation of the ellipse put y = 0; then the roots
of the roots of the resulting equation
Ax 2 + 2Gx + 1 =
will be the abscissae of the points of intersection of the ellipse and
the #axis. Representing these roots by xi and x%, we have
A = L, G = _^±^? (g)
X1X2 2x\Xz
Similarly, by putting x = 0, we obtain
B = J_, F =  y ^ (9)
2/12/2 2yij/ 2
These four constants are thus obtained by direct measurements
of the distances from the principal star to the intersection points
of the ellipse with the x and j/axes, in which care must be taken
to regard the algebraic signs. The fifth constant, H, is then
derived from the equation
H = Ax2 + B y 2 + 2Gx + 2p y + x do
2zi/
Measure the coordinates of several welldistributed points on
the apparent ellipse, so chosen as to make the product xy as large
as possible and substitute each set of values successively in
Eq. (10). The accordance of the resulting values of H will
depend upon the care with which the ellipse has been drawn,
and the mean of all should be adopted.
The values of the coefficients A, H . . . F being known,
we proceed as follows, adopting the analysis recently given by
W. M. Smart:*
Construct a sphere (Fig. 3) with the principal star S as center,
and let SL, SM, SK be the rectangular axes to which the general
equation of the apparent ellipse, in the plane of the great circle
LNM (at right angles to the line of sight), is referred, SL denning
the direction to position angle 0°. Let the great circle FNAB
define the plane of the true orbit relative to the primary. Then
QSN is the line of nodes, and, assuming the position angles to
increase from L toward N, LN is the longitude of the nodal
point £2, as denned in an earlier paragraph. Let P be the peri
astron point, and A the corresponding point on the sphere.
* W. M. Smart, On the Derivation of the Elements of a Visual Binary
Orbit by Kowalsky's Method, Mon. Not. R.A.S., 90, 534, 1930.
82
THE BINARY STARS
Then NA denotes co, and ANM, the inclination, i, taken to lie
between 0° and 90°.
Take rectangular axes SA, SB, and SC with reference to the
plane of the true orbit, and let (£, r], o) be the coordinates of
the companion at any time with respect to these axes. Since the
Fig. 3. — Diagram for Kowalsky's orbit method.
primary S is at a focus of the true orbit, the equation of the true
ellipse is
(£ + aeY
+ ?il
(11)
a 2 6 2
where 6 2 = a 2 (l — e 2 ).
Let (h, mi, ni), (Z 2 , m%, n 2 ), (£3, wi 3 , n 3 ) denote the direction
cosines of SA, SB, and SC, respectively, with reference to the
axes SL, SM, and SK. Then, drawing great circle arcs from
each of A, B, and C in turn to L, M, and K, we can write
Zi = cos AL Wi = cos AM, ni = cos AK,
l 2 = cos BL m 2 = cos BM, n 2 = cos BiiT,
Z 3 = cos CL, mi = cos CM, n 3 = cos CK,
and from the appropriate spherical triangles we derive at once
Zi = cos ft cos co — sin ft sin co cos £}
mi = sin ft cos co + cos ft sin co cos t> (12)
Wi = sin co sin i '
THE ORBIT OF A VISUAL BINARY STAR 83
l 2 = —cos 12 sin co — sin 12 cos co cos tl
m,2 = —sin 12 sin co + cos 12 cos co cos i> (13)
712 = cos co sin i J
U = sin 12 sin i 1
m 3 = —cos 12 sin i> (14)
w 3 = cos i )
We require subsequently the following relations between the
direction cosines:
Z1WI2 — hmi — n% (15)
h 2 + U 2 + Zs 2 = 1 (16)
ffH 2 + m 2 2 + m 3 2 = 1 (17)
Ixrrii + Z 2 w 2 + hm 3 — (18)
Any point (£, 1?) on the true orbit projects into the point (x, y)
on the apparent orbit. We have, consequently,
x = li% + Z 2 i?
y = mi£ + m 2 T?
from which, using Eq. (15),
.. m& — hy
? = z.
n 3
m\X — l\y
n 3
Substituting these values of £ and r\ in Eq. (11), we obtain
(m 2 x — hy + gens) 2 (mis  Ziy) 2 _  nQ v
sw + w u ;
This is the equation of the apparent orbit and is therefore equiva
lent to Eq. (4) (page 75). The coefficients of the same powers
of the variables in Eqs. (19) and (4) are therefore proportional.
Denoting the common ratio by /, we have, considering the coeffi
cients of x 2 , y 2 , . . . in turn,
A " tA a 2 & 2 /' ^ 2 \a 2 "*" 6 2 /i
H = Jjf 1 ^! + h^l) \ (20)
n ^A a 2 ^ 6 2 / ( K '
q _ fem 2 . p _ feh
ana ' an*
84 THE BINARY STARS
and, from the absolute terms,
1 = /(e 2  1)
or, writing
p = a(l  e 2 ) ■ £, (21)
/ =  (22)
This value of / can then be substituted in the expressions of
Eq. (20).
We now derive the elements in terms of A, B, . . . F. We
have, firstly, using Eqs. (21) and (22),
F 2  G 2 + A  B = e
p 2 n s 2
n* 2^ a / W  Z 2 2 Wl 2  Zx 2 \
(Z 2 2 — W 2 2 ) .1 5 T5 J
pn 3 2 \ a 2 b 2 )
= (*2 2 ~ m**) ( c , + 2) ( ffl ' 2 ~ W
p 2 n z 2 \ a/ p 2 n 3 2
= J,0i 8 + ^ 2  mx 2  m 2 2 )
= ^4?(^3 2  Z 3 2 ) by Eqs. (16) and (17).
Hence, using the values of h, ra 3 , n 3 given by Eq. (14), we obtain
F 2  G 2 + A  B = ^^ cos 212 (23)
Again,
FG — H = e ^ m ^ a_(yni ItmA
p 2 n s 2 pn 3 2 \ a 2 ap /
p 2 riz 2
_ Urrtz
p 2 n 3 2
by (18). Hence, using Eq. (14),
FG  H = Y^^ sin 212 (24)
From Eqs. (23) and (24) we obtain
(F 2  G 2 + A  B) sin 212 + 2(FG  H) cos 212 = (25)
which determines 12 in terms of known quantities.
THE ORBIT OF A VISUAL BINARY STAR 85
The value of tan 2 i/p 2 can then be found from Eq. (23) or (24).
Again, it is easily seen that
F 2 + G 2  (A + B) = £ 2 (!x 2 + h 2 + mi 2 + m 2 2 )
= ii(2  ^ 2  V)
1
,(2  sin 2 i)
P' COS" 6 t
= 2 W< (26)
P 2 P 2
Since we already have the value of tan 2 i/p 2 , Eq. (26) enables
us to calculate p, and therefore i.
Now, from Eqs. (13) and (20) we find
G'D COS %
— m 2 = sin ft sin co — cos ft cos co cos i = (27)
J5 T 'U COS %
— l 2 = cos ft sin a) + sin ft cos co cos t = (28)
Multiply Eqs. (27) and (28) by sin ft and cos ft, respectively,
and add. Then
e sin co = p(G sin ft — F cos ft) cos i (29)
Multiply Eqs. (27) and (28) by cos ft and sin ft, respectively,
and subtract. Then
e cos co = p(G cos ft 4 F sin ft) (30)
Hence,
(G sin ft — F cos ft) cos i ,__
tan CO = j~ , F . prr — (31)
(G cos ft + F sin ft) v
This equation, with Eqs. (29) and (30), determines a without
ambiguity.
The eccentricity e may now be obtained from Eq. (29) or
(30) and the semimajor axis a, from Eq. (21)
p = a(l  e 2 )
To complete the solution analytically, the period P and the time
of periastron passage T, are to be found from the mean anomalies
M , computed from the observations by taking the ephemeris
formulas on page 79 in reverse order. Every M will give an
equation of the form
86 THE BINARY STARS
M = ?~(t  T), or M = yi + e
where
e = nT.
From these equations the values of M and T are computed by
the method of least squares.
It is more convenient, however, to derive the values of P and T
from measures on the apparent ellipse made with the aid of a
planimeter, as follows: The diameter of the apparent ellipse
drawn through the origin S is obviously the projection of the
major axis of the true orbit, and the extremity of this diameter
nearest the origin is therefore the projection of the point of
periastron. Call it P'. Then, determine c, the constant of
areal velocity, from planimeter measures of the entire portion
of the ellipse covered by the observations, or, of that portion of
this ellipse which seems to be most accurately denned by the
observations; measure also the areas of two sectors P'Sp and
P'Sp', p and p' being two observed positions on opposite sides
of the point P'. Divide these areas by c and apply the quotients,
with proper signs, to the times corresponding to the positions
p and p' } and thus derive two values of T, the time of periastron
passage, which should agree closely. Several sets of points may,
of course, be used and the mean of all values for T adopted.
Similarly, the area of the entire ellipse may be measured, and the
result divided by c to find P, the revolution period. It is not
necessary to know the unit of area in making these measures,
since all the areas are simply relative.
ZWIERS' METHOD*
Zwiers' method is essentially graphical, and assumes that the
apparent orbit has been drawn. It may be well to insist again
that the utmost care must be exercised in drawing this ellipse,
for unless it gives a good geometrical representation of the obser
vations and satisfies the law of areas, it is useless to proceed with
the orbit computation.
The apparent ellipse being the projection of the true orbit, its
diameter drawn through S, the position of the principal star,
is the projection of the true major axis, and its conjugate, the
* A. N. 139, 369, 1896. Prof. H. N. Russell independently worked out a
method based upon the same geometric concept. A. J. 19, 9, 1898.
THE ORBIT OF A VISUAL BINARY STAR 87
projection of the true minor axis. Further, if P is that extremity
of the diameter through S which is nearest S it will be the projec
tion of the point of periastron passage in the true orbit. There
fore, letting C represent the center of the ellipse, the ratio
CS/CP will be the eccentricity, e, of that orbit, since ratios are
not changed by projection.
Let K = l/\/l — e 2 be the ratio of the major to the minor axis
in the true orbit; then, if all of the chords in this orbit parallel
to the minoi axis are increased in the ratio K:l, the ellipse will
be transformed into Kepler's eccentric circle. Consequently,
if in the apparent ellipse all ordinates parallel to the conjugate
diameter, described above, are prolonged in the ratio K:l we
shall have another conic which may be called the auxiliary
ellipse. It will evidently be the projection of the eccentric circle.
The major axis of the auxiliary ellipse will be a diameter of
the eccentric circle and therefore equal to the major axis of
the true orbit, and its position will define the line of nodes,
since the nodal line must be parallel to the only diameter not
shortened by projection. Designate the semimajor and semi
minor axes of the auxiliary ellipse by a and £, respectively;
then the ratio /3:a is the cosine of the inclination of the orbit
plane to the plane of projection. Again, the angle a' between
the major axis of the auxiliary ellipse and the diameter PSCP'
of the apparent orbit is the projection of the angle w, the angle
between node and periastron in the true orbit. Therefore
tan o)' a , /OON
tan o> = r = — tan co (32)
cos i j8
Finally P and T are found by areal measures in the apparent
ellipse in the manner already described.
The conjugate diameter required in Zwiers' construction
may be found most easily by first drawing any chord of the
ellipse parallel to PSCP', the projected major axis. The
diameter through the middle point of this chord is the con
jugate required. If desired, advantage may also be taken of
the fact that the conjugate diameter is parallel to the tangents
to the ellipse at the points P and P', and the rectangular axes
of the auxiliary ellipse found by trial or by the following con
struction : Let
x 2 . y 2 1
(a') 2 "•" (&') 2
88 THE BINARY STARS
be the equation of the apparent ellipse referred to its conjugate
diameters. The equation of the auxiliary ellipse referred to
the same axes will be
x z y 2 _
(a 7 ) 5 + {Kb'f ~
The axes are therefore also conjugate diameters of the auxiliary
ellipse. At the extremity P of the diameter a' (PSCP'), erect
two perpendiculars, PA and PB, to the tangent to the ellipse
at this point and make each equal in length to Kb'. Through
the extremities of the two perpendiculars and the center C of
the apparent ellipse pass a circle. It will cut the tangent in
two points, A' and B'. The lines A'C t and B'C will give the
directions of the two rectangular axes required, the major
axis lying in the acute, the minor axis in the obtuse angle between
the diameters a' and Kb'.
Instead of actually constructing the auxiliary ellipse it will
generally be easier to derive the elements directly from meas
ures of the apparent ellipse with the aid of simple formulas
based upon the analytical solution of the construction. Thus:
Let e, a', and b' again represent, respectively, the eccentricity,
and the projected major and minor axes of the orbit, and let
x\ and X2 be the position angles of a' and b r . To avoid ambig
uity, let Xi be the position angle of the principal star as viewed
from the center of the apparent ellipse and let X2 be so taken
that (xi — X2) is an acute angle. Also, compute as before,
K = 1/V1 — e 2 and b" = Kb'. Then the relations between the
rectangular axes 2<x and 2/3 of the auxiliary ellipse and the con
jugate diameters 2a' and 2b" are given by the equations
a 2 + £2 = fl /2 __ & »2
a($ = a'b" sin {x x  x 2 ) (33)
the sine being considered positive.
The coordinates of any point on the auxiliary ellipse with
respect to the axes 2a and 2/3 may be written in the form
a cos 4>', 18 sin 4/. Let a cos (w), £ sin (w) be the coordinates of
the extremity of the a' diameter; then we shall have
a' 2 = a 2 cos 2 («) + 2 sin 2 (co) (34)
and
la 2 — a' 2
tan (co) = ±^^—j2 < 35 >
THE ORBIT OF A VISUAL BINARY STAR 89
in which the sign of tan (co) is to be the same as that of (xi — x 2 ).
But co', the projection of w is related to (co) by the equation
tan co' =  tan (co) (36)
a
that is («) = co and ft = (xi — co').
The angle co obviously may have either of two values differ
ing by 180°; that value is to be taken which will make ft less
than 180°.
Zwiers counts all angles in these formulas in the direction of
increasing position angles.
The practical procedure may therefore be stated as follows:
Construct the apparent ellipse and the diameter b' conjugate
to a'\ measure e, a', V, xi and x 2 ; compute K = l/\/l — e 2 >
b" = Kb', and find a and @ from
(a ± j8) 2 = a' 2 + 6" 2 ± 2a'b" sin (xi  X*)
the sine being taken positive. Then
a = a
•
cos % = 
a
, / « 2  a' 2
tan co = ±yj a ,i_p >
the sign of tan co being taken the same as that of (xi — z 2 ), and
of the two values of co that one which makes S2 less than 180°.
Next we have
tan <a' =  tan co £2 = («i — co'),
a
and finally deduce the values of P and T from area measurement,
as in the GlasenappKowalsky method.
THE THIELE METHOD AND THE THIELEINNES CONSTANTS
In 1883,* T. N. Thiele published a method of orbit computa
tion depending upon three observed positions and the constant
of areal velocity, and substituting for the elements a, i, co, and ft,
the polar coordinates (a, A) and (6, B), referred to the center
of the apparent ellipse as origin, of two points, P' and R', the
projections upon the apparent ellipse of the points P and R in
* A. N. 104, 245, 1883.
90 THE BINARY STARS
the true orbit having, respectively, the eccentric anomalies
0° and 90°. The three observed positions should, of course, be
normal places carefully selected to represent as long an arc as
convenient of the wellobserved part of the apparent ellipse,
for the success of the computation depends upon the skill with
which this selection is made, and the accuracy with which the
areal constant, defined later, is determined.
Although the method has a wide range of applicability, it
did not come in to extensive use until quite recently, when it
was revived by Innes and van den Bos. Innes, * in 1926, seeking
a simpler method than those in common use for correcting the
preliminary elements of an orbit differentially, independently
developed a method of orbit computation which differs from
Thiele's only in that he used rectangular instead of polar coordi
nates and for Thiele's points R and R' substituted Q, the point
on the auxiliary circle drawn on the major axis of the true orbit
that corresponds to R and Q' its projection on the plane of the
apparent ellipse. If A, B, F, and (?, represent these coordinates,
the relations between them and Thiele's are :
Innes Thiele
A a cos A
B a sin A
F b sec <p cos B
6 b sec <p sin B
where sin <p — e.
It will be convenient to treat the two methods together, and
to follow Innes and van den Bosf rather than Thiele in the
development of the formulas.
The data assumed as known are the three normal places
corresponding to the times ti, U, h and C, the double constant of
areal velocity. Each observation gives
= p cos d\
= p sin df
(37)
These are connected with the usual orbit elements by
p cos (0 — Q) = r cos (v + w) ) , .
p sin (0 — Q) == r sin (v + «) cos i)
* W. H. van den Bos, Orbital Elements of Double Stars, Union Obs., Circ.
68, 354, 1926; 86, 261, 1932.
fLoc. cit.
THE ORBIT OF A VISUAL BINARY STAR 91
Removing 12 to the righthand number, and separating v from co,
we have
(39)
x = r cos v (cos w cos 12 — sin co sin 12 cos i) +
r sin t; (—sin co cos $2 — cos co sin 12 cos i) i
2/ = r cos w (cos co sin 12 + sin co cos 12 cos i) +
r sin t> (—sin co sin 12 + cos co cos 12 cos i) t
It is to be noted that in all the ThieleInnes formulas the
inclination i, for retrograde motion, is taken between 90° and 180°.
Put
r
X —  cos v = cos E — sin <p\
a '
r
Y =  sin v = cos «> sin 2?
a
where sin <p = e
and
A = a (cos co cos Q — sin co sin $2 cos i)
B = a (cos co sin + sin co cos 12 cos i)
F = a ( — sin co cos 12 — cos co sin U cos i) I
(r = a ( — sin co sin 12 + cos co cos 12 cos i) ,
Then we have
x = AX + FF)
y = 5X + GYf
(40)
(41)
(42)
Equations (41) show the relations between Innes* constants and
the elements a, i, co and 12, while Eqs. (42) are his fundamental
equations. It is also clear that the points (A, B), (F cos v,
G cos <p), with the center of the apparent ellipse, define a pair of
conjugate axes which are the projections of the major and
minor axes of the true orbit.
The double areal constant is, obviously, twice the area of the
entire ellipse divided by the period. Since the area of an
ellipse is t times the product of its semiaxes, and the area of the
rectangle constructed on the axes equals the area of the parallelo
gram constructed on any pair of conjugate axes, we have
C = ^(AG  BF) cos <p = n(AG  BF) cos <p (43)
where p = 2t/P — 6.28319/P is the mean annual motion
expressed in radians. Let A p , q represent the double area of
92 THE BINARY STARS
the triangle formed by the position of the primary and the
positions of the companion at the times t p and t q . Then
Ap.a = x p y q  x q y p = (AG — BF)(X p Y q — X q Y P ),
or, introducing values from Eqs. (40) and (43),
^ =  [sin {E q  E p )  sin ^(sin E q  sin E p )] (44)
From Kepler's equation, E — sin (p sin E = M = n(t — T),
we find
t q t p = [(Eq ~ E P)  Sil1 ^( Sil1 E 1 ~ Sil1 E P)\
A 1
ana, subtracting Eq. (44), we obtain
t q  t p  ^1 = I [(E q  E p )  sin (E q  E p )] (45)
This, in different notation, is Thiele's fundamental equation.
Let us put E 2  Ei = u, E 3  E 2 = v, E z  E x = u + v,
and write Eq. (45) for (h — ti), (t z  h), and (t z — h). We
have
h — h jr =  (u — sin u)
hh—^.=(v sin v) \ (45a)
U  h  ^ =  [(u +v)  sin (u + v)]'
The left hand members of the three equations are known from
the three normal places and the double areal constant.
If the observations extend over a full revolution, so that the
period is approximately known, the approximate value of m
will be known and we may proceed at once to derive the values
of e (= sin <p) and E*. If this is not the case, we assume the
most plausible value for /x, and, with the help of a table for
(x — sin x), compute u, v and (u + v). By successive approxi
mations a value for n should readily be found which will make the
values of u and v check with the value of (u + v). Then
D 2x 6.28319 , , 70fl , c
P = — = , and n = 57.2958m
THE ORBIT OF A VISUAL BINARY STAR 93
From Eq. (44) we have, by introducing u, v, (u + v), Ai, 2 ,
A 2 ,3 and A 3 ,i successively, and combining,
Q
Ai,2 + A 2 ,3 — Ai,3 = —[sin u + sin v — sin (u + v)]
/*
and
Q
A 2 ,3 sin u — Ai, 2 sin v = — sin p[sin v(sin E 2 — sin E x ) —
M
sin w(sin E% — sin E 2 )\.
Substituting (2£ 2 — J^i) and (Ez — E 2 ) for u and t> in the right
hand member of this equation, removing all parentheses, and
recombining and substituting, we finally derive
Q
A 2>3 sin u — Ai, 2 sin v = — sin <p sin E% [sin u + sin v — sin (u + »)]
(46)
Similarly, from
A 2 ,3 cos u + Ai, 2 cos v — Ai,3 = — sin p[sin Ez — sin Ex
V
— cos w(sin Ez — sm E 3 — cos t>(sin 2? 2 — sin E x )\
we derive
A 2 ,3 cos w + Ai, 2 cos v — Ai,3 = — sin <p cos E 2
[sin w + sin v — sin (w + v)] (47)
and from Eqs. (46) and (47)
. „ A 2 ,3 sin u — Ai, 2 sin v
sm <p sin E 2 = — 2 — : —
Ai, 2 + A 2>3  A 1>3
«;« „~<, c A 2(3 cos w + Ai, 2 cos v Ai. s ( *■ '
sin ^> COS /i 2 = r ; t r
Ai, 2 + A 2>3 — A x .3
These equations give E 2 and e = sin ^; then E\, E 3 follow from
(E 2 — u) and (E 2 + v). For each time, h, t 2 , t 3 , we next derive
the mean anomaly from Kepler's equation and thus three values
of T, the time of periastron passage, which should agree closely.
The values for X and Y follow from Eq. (40) and, finally, those
of the four constants, A, F, B, G, from the normal places by the
use of Eqs. (42).
94 THE BINARY STARS
Whatever method of orbit computation is adopted, it is recom
mended that the ThieleInnes constants as well as the elements
in the usual notation be given. The formulas, to convert from
the one system to the other, in convenient form for logarithmic
computation are:
To derive the ThieleInnes constants
A + G = 2a cos (a> + ft) cos 2  ]
z
A — G = 2a cos (w — ft) sin 2 I
& *
B  F = 2a sin (w + ft) cos 2 ^L
B  F = 2a sin (o>  ft) sin 2 ^]
For the inverse process,
D f
tan (<o + Q) = A . g
tan (co  ft) = ~f ~I } (MY
(49)
tan
A G
i AG cos(co + G) BF sin(oj + fi)
2 A+G cos (cofi) BF sin (« ft)
It may again be pointed out that in these formulas *', in the case of
retrograde motion, is taken in the second quadrant.
The practical procedure, then, is first to form the three normal
places and find the value of C the double areal constant. Thiele
employs processes of numerical integration for this purpose,
but it is better, as well as more convenient, to utilize the carefully
drawn apparent ellipse, or interpolating curves for position angles
and distances against the times. It is hardly necessary to say
that the normal places must conform to the law of areas, and,
that, if they are derived from interpolating curves, care must
be taken that they fall upon the curve of the apparent ellipse;
but it may be well to note that C ( = p 2 ^ j has the negative sign
when the position angles decrease with the time and that the
units for p and dd/dt are, respectively, seconds of arc, and radians
per annum.
* [The tan 2  formulas are awkward. A better method of solution for a, i,
to, and CI, is:
a(l + cosi) = (A + G) sec (co +0) = (B  F) esc (w +0)
a(l  cosi) = (A  <?)sec(to Q) = (B  F) esc (o> Q),
from which a and i may be computed easily. — J.T.K.]
THE ORBIT OF A VISUAL BINARY STAR
95
Having the required data, we first find P through the value of n
derived from Eq. (45a), then e and T, from Eq. (48) and Kepler's
equation, and finally X and Y and the four constants A, B, F, G,
from Eqs. (40) and (42). In other words, the three purely
dynamic elements are derived first, and then values of the four
constants (representing the geometric elements) to correspond.
ILLUSTRATIVE EXAMPLES
In the first edition of this book, I used my computations of the
orbit of A88 (ADS, 11520) to illustrate the GlasenappKowalsky
180'
Fig. 4. — The apparent orbit of A 88. The solidly drawn broken line connects
the positions used in the computation of the orbit; the dashed and dotted
lines connect the positions given by later measures.
method and the method of Zwiers. The period of the system
is 12.12 years, and the observations now available show that the
companion has described more than 23^ revolutions about the
primary since discovery. These later observations, when com
pared with an ephemeris based on the orbit, give residuals
comparable to those obtained for the observations on which the
orbit was based. I have therefore not revised the elements,
but retain the original computations which were made in 1912,
96
THE BINARY STARS
Measures and Residuals for A88
Date
1900.46
1901.56
1902.66
1903.40
1904.52
1905.53
1906.48
1907.30
1908.39
1909.67
1910.56
1911.55
1912.57
1914.55
1914.55
1915.52
1916.24
1916.63
1916.76
1917.62
1917.64
1918.52
1918.76
1919.62
1920.37
1920.67
1921.52
1921.53
1922.62
1923.57
1923.76
1924.51
1924.65
1925.61
1928.63
1931.66
1932.78
1933.60
3532
338.3
318.1
293.6
278.4
224.8
199.1
193.5
178.1
150.4
47.0
18.7
356.1
331.2
330.2
306.4
277.2
243.0
248.8
222.5
228.1
200.4
196.9
188.4
173.6
172.6
143.5
144.4
Too close
10.9
11.8
354.9
344.2
340.6
272.2
187.5
177.7
159.3
Po
O'.'H
0.14
0.12
0.11
0.14
0.12
0.13
0.14
0.15
0.10
0.11
0.15
0.15
0.14
<0.20
0.15
0.13
0.16
0.14
0.10
0.14
0.14
0.14
0.15
0.16
0.16
0.15
0.12
0.14
0.18
0.15
0.12
0.15
0.14
0.14
0.15
0.11
Obs.
Rabe
VBs
Lv
VBs
VBs
Btz
BtF
Mag
VBs
(OC)
Ad
 13
 0.8
+ 0.7
 0.2
+20.8
 0.2
 7.4
 0.8
+ 0.1
+ 3.3
 0.6
+ 7.0
+ 0.9
+
+
9.2
8.2
+ 10.6
 9.8
12.1
 2.3
 3.6
2.
7.
7.
3
+
 7.4
 2.6
 6.9
 6.0
 2.9
+ 1.9
 2.9
11.6
 1.5
+ 12.8
 6.9
+ 2.4
+ 6.6
0''004
0002
+0.002
0.000
+0.040
0.000
0.020
0.002
+0.000
+0.006
0.001
+0.017
+0.003
+0.022
+0.020
+0.020
0.017
0.023
0.004
0.008
+0.008
0.021
0.020
0.009
0.020
0.007
0.011
0.009
0.007
+0.005
0.009
0.034
0.004
+0.025
0.020
+0.006
+0.012
A P
0'.'03
0.02
0.01
0.00
+0.03
+0.01
0.03
0.03
0.00
+0.01
+0.03
+0.01
0.02
0.00
+0.04
+0.03
+0.05
+0.03
+0.03
+0.01
0.02
0.02
0.02
0.00
+0.01
+0.06
+0.03
0.00
+0.03
0.02
0.05
0.01
+0.03
0.03
0.00
+0.01
The average residual in angle expressed in radians, A0(p/57.3), is ±0''012; and in dis
tance, Ap, ±0'/022.
THE ORBIT OF A VISUAL BINARY STAR 97
adding, however, the later measures and the corresponding
residuals.
All of the measures to date are given in the table on page 96.
The dates, observed position angles, and observed distances are
recorded in the first three columns. The fourth column shows
the number of measures (on different nights) on which each posi
tion rests. Almost all of the measures were made by me, with
the 36in. refractor; for the others, the name or an abbreviation
of the name of the observer is entered in the fifth column.
(VBs = Van Biesbroeck, Lv = Leavenworth, Btz = Bernewitz,
Mag = Maggini.) The last three columns give the residuals,
observed minus computed, for the position angles and distances,
the former entered both in degrees and reduced to stricter
comparison with the distance residuals by multiplying by the
factor p/57.3.
All of the measures to 1912 inclusive were plotted, using a
scale of 3 in. to 0"1, and, after repeated trials, the ellipse shown
in the diagram on page 95 was drawn. It represents the observa
tion points fairly and satisfies the law of areas closely. Applying
the GlasenappKowalsky method, we first measure the intercepts
of the ellipse with the axes of coordinates, and the coordinates of
two selected points for the value of H, counting the end of the
a>axis at 0°, and of the t/axis at 90°, positive. The measures
are (in inches on the original drawing) :
xi = +4.98, ?/i = +1.77, x a = 2.55, y a = 2.86
Xi = 4.73, y* = 3.12. x b = +3.17, y b = 2.49
Therefore we have
xix 2 = 23.5554, y x y 2 = 5.5224, x a y a = +7.2930,
xi + x 2 = +0.25, yi + y 2 = 1.35, x b y h = 7.8933,
xj = 6.5025, y a 2 = 8.1796
Xb 2 = 10.0489, y b 2 = 6.2001
from which to compute the five constants of the equation of
the ellipse. We find
A = — = 0.04245
#1#2
B = — = 0.18108
F =  y \ + y * = 0.12223
22/i2/2
98 THE BINARY STARS
G = _ *i+ x * = +0.00531
2zi£ 2
From these values and the coordinates x a , y a , we obtain
H =  Ax * + B * + ™ x + ^ + 1 = +0.00584,
2xy
and, similarly, from the coordinates Xb, yb,
H = +0.00590,
and adopt the mean, +0.00587.
Combining these constants, we have,
FG = 0.00065; F 2 = +0.01494; G 2 = +0.00003;
2(FG  H) = +0.01304; F 2  G 2 + A  B = +0.15354;
F 2 + G 2  (A + B) = +0.23850.
The solution of Eqs. (24), (23), (26), (29), (30), and (21)
then proceeds as follows :
log i ^l sin 2fl 8.11528
p 2
log i ^J cos 212 9.18622
log tan 2S1 8.92906
2 fl 4?85
fi 2?42
log cos 212 9.99844
log ***£* 9.18778
tan 2 *
+0.15409
9 ton 2 i"
From Eq. (26)  2 + ^ +0.23850
2
J.
»2
+0.08441
+0.04220
log  a 8.62536
log p 2 1.37464
log v 0.68732
log tan 2 1 0.56242
log tan i 0.28121
180°  i = 62°22'30" = 62 °A
THE ORBIT OF A VISUAL BINARY STAR
99
log
logF
sin 12
cos 12
(1) log F cos 12
(2) log F sin $2
(1)
(2)
(3)  (1)
log [(3)  (1)]
cos i
logp
log e sin u
log tan co
CO
sin a
log e
e
e 2
1  e 2
log (1  e 2 )
logp
V
log
9.08718 n
8.62557
9.99962
9.08680 n
7.71275 n
0.12212
0.005161
+0.12234
9.08757
9.66586
0.68732
9.44075
2.59507*
90?1
0.00000
9.44075
0.276
0.07618
0.92382
9.96559
0.68732
(1  e>)
= log a 0.72173
a 5.269 in.
= 0^176
i 117?6
cos i 9.66586
logG
sin J2
cos $2
(3) log G sin J2
(4) log G cos 12
(3)
(4)
(2) + (4)
log  [(2) + (4)]
V
7.72509
8.62557
9.99962
6.35066
7.72470
+0.00022
+0.005305
+0.000144
6.15836„
0.68732
log e cos co 6.84568n
From the diagram. (Fig. 4) it is obvious that the companion
passed its periastron point between the dates of observation
1909.67 and 1910.56; but the measures made in 1908 and 1912
were regarded as more reliable than these and were accordingly
used to determine the time of periastron passage. The constant
of areal velocity (in units of the planimeter scale) had been found
to be 0.205. Drawing radii to the points P and 1908.39 and
1912.57, the areas of the two resulting sectors were, in terms
of the same unit, respectively, 0.34440 and 0.50225. Hence
the time intervals between these two dates and the date of
periastron passage were, respectively, +1.68 years, and —2.45
years, giving for T, the two values, 1910.07 and 1910.12. The
mean, 1910.1, was adopted. The planimeter measures gave as
100 THE BINARY STARS
the area of the entire ellipse, 2.4848, and the period, 12.12
years.
To solve the orbit by Zwiers' method, we begin by finding
the axis b' conjugate to a' (PSCP'). Draw the chord cc par
allel to P'CSP and then draw the diameter through its middle
point. This will be the required conjugate.
We now measure CS = 0.67, CP = 2.45, a' = 2.445, b' = 5.050;
and the angles Xi = 92°6 and x 2 = 3°6.
The ratio CS:CP gives at once the value of the eccentricity,
e = 0.273, and from this we compute the value of K = \/\/l — e 2
(in logarithms) 0.01682. Thence we find b" = Kb' = 5.2494.
The computation then proceeds as follows:
log a'
0.38828
log b"
0.72011
log 2
0.30103
log sin (xi — X2)
9.99993
log 2a'b" sin (xi — x 2 )
1.40935
2a'b" sin (x% — x 2 )
25.6653
(a') 2 + (b"Y
33.5342
(a + £> 2
59.1995
(a " PY
7.8689
(a+/8)
7.6942
(«/8)
2.8052
2a
10.4994
20
4.8890
a
5.2497
P
2.4445
log/3
0.38819
log a
0.72014
log cos i
9.66805
.'. i =
62°25
a = a =
5.25 in.
(tf'175
(a')«
5.9780
(6") 2
27.5562
[(a') 2 + (6") 2 ]
33.5342
a 2
27.5600
2
5.9756
a 2  (a') 2
21.5820
THE ORBIT OF A VISUAL BINARY STAR 101
(«') 2  p
0.0024
log [a 2  (a') 2 ]
1.33409
log [(a') 2  J8 2 ]
7.38021
log tan 2 w
3.95388
log tan w
1.97694
.'. u =
= 89?4
log cos i
9.66805
log tan «'
1.64499
«'
88^7
/. = (xi  «') = 3?9
Assembling the elements we have the following :
Glasenapp's Method Zwiers' Method
P= 12.12 years
12 . 12 years
T = 1910.10
1910.10
e = 0.276
0.273
a = 0"176
, . , 175
w = 269^9
270^6
i = 117?6
117?75
Q = 2.4
3.9
Angles decreasing with the time.
In the formulas, all angles are counted in the direction of
increasing position angles, whereas in the notation given on
page 78, « is counted from node to periastron in the direction
of motion of the companion. Therefore, when as in this system
the. observed position angles decrease with advancing time, the
value for « derived from the formulae must be subtracted from
360°. In applying the formulas for computing the ephemeris
of such a system, the anomalies are counted positive after
periastron passage and negative before, just as in the case of
direct motion (angles increasing with the time) ; cos i is counted
as positive, and the angles (0 — fi) are taken in the quadrant
360° — (v + co). I have found this to be the simplest and most
satisfactory method of procedure in every case where the angles
decrease with the time. In orbits with direct motion the value
of « is used as given directly by the formulas and the angles
(6 — fl) are taken in the same quadrant as the angles (« + «).
The orbit of 24 Aquarii* has been selected as an illustration of
Thiele's method, as it is a beautiful example of the power of that
*24 Aqr = B 1212 = BDS 11125 = ADS 15176.
102
THE BINARY STARS
method in dealing with a rather recalcitrant case. The orbit
was computed by W. S. Finsen and is published in Union Observa
tory Circular 81, 112, 1929. In the list of measures on page
103 the columns give respectively the date, the position angle, the
distance, the number of nights, the aperture, the observer, and
the residuals (observed minus computed) in angle and in distance.
An asterisk attached to the date denotes measures overlooked
in Finsen's compilation or not available at the time.
1930
Scale
Fig. 5. — The apparent orbit of 24 Aquarii.
The angles and distances were plotted against the time and
interpolation curves drawn. In the years 19221924 the meas
ures are discordant and uncertain; they were therefore dis
regarded, the areal constant being based on the arc 18911918
alone and the three normal places required being taken suffi
ciently far away from the unreliable part of the interpolation
curves.
The angles and distances were read for every second year as
shown in the table on page 104.
The numbers in the column headed did 2 Ap, giving the products
of two successive distances and the sectorangle between them,
may be considered sufficiently close approximations to the double
areas of the sectors (provided that the distance does not vary too
greatly within the sector) and should therefore be constant.
THE ORBIT OF A VISUAL BINARY STAR
Measures and Residuals for 24 Aquarii
103
Tel.
Obs.
O 
 C
Date
00
PO
n
A9
Ap
1890.75
254?5
0745
3
36
 2.0
0708
1.75
261.0
0.55
4
36
+ 4.0
+0.01
2.40
256.2
0.38
2
19
Sp
 3.0
0.16
3.68
260.5
0.55
3
16
HCW
 0.6
0.00
3.88
262.8
0.59
1
36
Bar
+ 1.4
+0.04
4.82
264.7
0.52
7
36
Bar
+ 1.9
0.03
4.86
261.5
0.45
3
19
Sp
 1.3
0.10
7.81
263.5
0.65
3
36
A
 3.6
+0.09
7.89
267.4
0.73
1
26
SBn
+ 0.2
+0.17
8.78
269.0
0.49
3
36, 12
A
+ 1.7
0.07
8.84
269.0
0.54
1
40
+ 0.4
0.02
9.98
282.6
0.5e
1
26
SBn
1900.67
205.1
0.5e
1
6
Sola
0.74
271.9
0.55
2
14
Dob
+ 0.7
0.01
1.54
269.4
0.49
10
28
GrO
 3.0
0.07
1.79
274.0
0.55
2
36
A
+ 1.2
0.01
2.00
273.0
0.57
11
18
Doo
 0.1
+0.01
3.86
273.8
0.54
2
15
VBs
 1.9
0.01
4.54*
273.0
0.48
5
15M
Com
 3.7
0.07
4.67
278.6
0.49
1
36
A
+ 1.7
0.06
8.35
282.6
0.52
3
15K
Com
 0.1
0.00
8.47
285.5
0.49
4
15
VBs
+ 2.7
0.03
8.72
279.6
0.68
2
26
Ol
 3.6
+0.16
8.72
284.8
0.56
2
26
RW
+ 1.6
+0.04
8.73
286.4
0.72
2
26
Neff
+ 3.2
+0.20
9.85
281.1
0.58
3
18
Doo
 4.1
+0.07
1910.40
282.2
0.51
3
15H
Com
 3.9
+0.01
0.72
278.2
0.43
5
28
GrO
 8.5
0.07
1.20
291.9
0.52
3
6
Dob
+ 4.3
+0.02
1.68
286.8
0.50
3
15
VBs
 1.6
+0.01
4.00
292.5
0.47
8
28
GrO
 0.5
+0.01
4.63
291.3
0.47
2
36
A
 2.7
+0.02
4.65
294.6
0.48
2
15^
Com
+ 0.6
+0.03
4.66*
293.5
0.51
1
40
Lv
 0.5
+0.06
4.79*
298.1
0.45
3
10M
Lv
+ 3.6
0.00
4.94
311.1
0.41
5
8
Rabe
+ 16.0
0.04
6.42
296.5
0.53
3
26
Ol
 2.0
+0.11
6.62
293.1
0.46
3
12,40
Lv
 5.9
+0.04
6.65
301.9
0.41
3
15H
Com
+ 2.9
0.01
7.69
305.9
0.40
3
15J^
Com
+ 3.9
0.00
7.74
294.7
0.42
1
40
VBs
 7.5
+0.01
1921.66
321.1
0.22
3
36
A
+ 3.3
0.07
2.81
40 ?
<0.1
1
36
A
3.62
341.8
0.20
2
40
VBs
+ 11.3
0.01
3.85
6.0
0.57
4,3
30
Bail
3.88
333.5
0.15
3
13
Mag
+ 0.3
0.05
4.55
55
0.12
1
36
A
0.04
4.71
6.9
0.22
1
30
Plq
+0.07
4.82
350.0
0.16
1
40
VBs
+ 2.1
+0.02
1926.64
190.7
0.20
1
2616
B
10.0
+0.05
6.69
204.2
0.19
1
36
A
+ 2.9
+0.03
7.74
2H.0
0.21
3
26M
B
 3.6
0.02
7.74
218.7
0.23
1
26H
<P
+ 4.1
0.00
8.73*
224.6
0.26
1
36
A
+ 2.9
0.01
8.75
221.2
0.27
4
26M
B
 0.6
0.00
8.75
222.6
0.26
4
26H
<p
+ 0.8
0.01
9.46*
228.8
0.28
4
23>£
V
+ 3.1
0.02
9.63*
230.2
0.27
1
36
A
+ 3.8
0.04
9.86*
227.8
0.26
3
26W
B
+ 0.3
0.05
1930.48*
234.3
0.29
3
23>
V
+ 4.3
0.04
1.66*
236.0
0.37
2
36
A
+ 1.6
0.00
2.79*
236.9
0.35
4
26K
B
 0.9
0.04
2.79*
238.3
0.30
1
2GM
<p
+ 0.5
0.09
The actual figures show the need of further adjustment. As
this can be made both for Ap and for d, it is advisable to plot the
polar coordinates as a check that the adjusted positions show a
104
THE BINARY STARS
smooth elliptic arc. In general it is found that the major part
of the adjustment has to be made in the distances.
The columns on the right of the double rule show the result of
the adjustment, which may be regarded as satisfactory.
t
V
Ap
d
didzAp
V
Ap
d
d\.d%Ap
1890
255?5
+3?1
0'.'47
+0.71
255?4
+3?2
0"53
+0.92
2
258.6
2.9
0.49
0.74
258.6
3.0
0.54
0.89
4
261.5
2.5
0.52
0.70
261.6
2.9
0.55
0.89
6
264.0
2.0
0.54
0.59
264.5
2.8
0.56
0.89
8
266.0
3.0
0.55
0.92
267.3
2.7
0.57
0.88
1900
269.0
3.0
0.56
0.94
270.0
2.8
0.57
0.91
2
272.0
3.6
0.56
1.15
272.8
2.9
0.57
0.92
4
275.6
2.6
0.57
0.85
275.7
3.0
0.56
0.92
6
278.2
3.5
0.57
1.14
278.7
3.2
0.55
0.93
8
281.7
3.6
0.57
1.11
281.9
3.4
0.53
0.92
10
285.3
3.0
0.54
0.84
285.3
3.6
0.51
0.90
12
288.3
3.9
0.52
0.99
288.9
4.0
0.49
0.90
14
292.2
4.8
0.49
1.06
292.9
4.6
0.46
0.91
16
297.0
6.1
0.45
1.10
297.5
5.5
0.43
0.92
18
303.1
0.40
Mean
303.0
0.39
Mean
+0.917
+0.907
The mean value of did 2 Ap, +0.907, has now to be divided by
the interval in time, here 2, and by 57.3 to reduce Ap to radians.
We obtain for the double areal constant: c = +0.007914.
As the result of a similar adjustment Finsen found c = +0.00781
which he supplemented by the three normal places
THE ORBIT OF A VISUAL BINARY STAR 105
1892.00
258? 6
1910.00
285? 3
1928.00
21 T
0'54
0"51
0"24
The above adjustment, which is equivalent to, though experts
in the method regard it as considerably simpler than, the con
struction of the apparent orbit in the graphical methods, has been
given rather fully, for, as noted in describing the method, it is
the crux of the problem. It is at this stage that the computer's
judgment, experience, and knowledge of the reliability of the
heterogeneous observational material come into play; once
the three normal places and the double areal constant have been
found, the derivation of the orbital elements is simple and
straightforward.
It sometimes happens that the period is known a priori
(from the recurrence of position angle) while the character of the
apparent orbit, or large gaps in the observed arc, cause difficulties
in the determination of c. It is therefore worth remembering
that in Thiele's fundamental formula
t q  t P  — =  [E q E p sw (E q  E p )]
we may regard either /torcas unknown.
Adopting Finsen's normal places and c so as to obtain his
elements, the computation proceeds as follows:
U = 1892.00 xi = 0.107 y x = 0.529 A w = +0.1240
t 2 = 1910.00 x 2 = +0.135 y 2 = 0.492 A 2t3 = 0.1139
t 3 = 1928.00 x 3 = 0.192 y z = 0.144 Ai., = 0.0862
^ = +15.88 u  sin u = 2.12**
c
^ = 14.58 t;  sin v = 32.58/t
c
^ = 11.04 (u + v)  sin(w + v) = 47.04m
c
In the approximations on page 106 for /* the table for x —
sin x given in Union Observatory Circular No. 86 is used; it is
helpful to plot the differences found in the last row against the
values of n in the first row, as after three approximations have
been made the resulting differences frequently define the curve
so well, that the fourth approximation becomes final.
106
THE BINARY STARS
As we expect the period to be of the order of 50 years (/x = 0. 126) ,
we try n = 0.12 and /* = 0.13:
u — sin u
v — sin v
(i* + v ) — sin (u + v) .
u
v
Sum
(u +»)...
Difference.
0.12
0.2544
3.9096
5.6448
67.51
202.27
269.78
266.25
+3.53
0.13
0.2756
4.2354
6.1152
69.43
212.17
281.60
301.54
19.94
0.122 0.1224
0.
3.
5.
67.
204.
272.
271.
+0.
2586
9748
7389
90
22
12
53
59
0.2595
3.9878
5.7577
67.98
204.61
272.59
272.66
0.07
and find
M = 0.1224 u = 67?98 v = 204?61
Finsen, working to another decimal place, obtained :
H = 0.12241 u = 68?022 v = 204?642 Final difference
0?002
and hence for period and mean motion:
P = 51.33 years n = 7?0134
Further
e sin E 2 = 0.5598 e cos E 2 = 0.7178
e = 0.9102 E 2 = 217?95
tf i = E 2  u = 149?93 E 3 = E 2 + v = 62?59
{Mi 123?80
M 2 250?02
M 3 16?29
all giving T = 1925.68.
X x = 1.7754 Y t = +0.2078
X 2 = 1.6985 F 2 = 0.2550
X 3 = 0.4492 Y 3 = +0.3679
From the first and third normal places
A = O'.'OOll, B = +0'. , 2939, F = 0 , . , 5233, G = 0'.'0317.
These values give
x 2 = +0'.'135 y 2 = O'.^Ol
for the second normal place, a satisfactory check.
THE ORBIT OF A VISUAL BINARY STAR 107
Though the conflicting results obtained in the years 19221924
were entirely disregarded in deriving the orbit, partly because
it was not even possible to assign the quadrants with certainty,
the residuals show that the measures by Aitken in 1921, Maggini
in 1923, and van Biesbroeck in 1923 and 1924 are satisfactorily
represented. This success of the orbit in bridging the large gap
from 300° to 200°, a sector of 260° in all, inspires confidence in
the elements found.
A = 0.0011 B = +0.2939
G = 0.0317 F = 0.5233
A + G = 0.0328 B F = +0.8172
A  G = +0.0306 B F = +0.2294
cos (co + 12) negative, sine positive
cos (co — 12) positive, sine positive
I q gi 79
tan (co + 12) = _ ' 2g = 24.9 second quadrant
tan <  a >  £nfs  + 7  49 fir8t "
co + Q = 92?30
to  12 = 82.40
co = 87?35 J2 = 4?95 2co = 174.70
212 = 9.90
If 12 were found to be in the third or fourth quadrant, 180°
should be added to (or subtracted from) both co and 12.
cos (co + 12) = 0.0401 sin (co + 12) = +0.9992
cos (co  12) = +0.1323 sin (co  12) = +0.9912
x 9 i +0.2294 +0.9992 OQQn i , nW9n
tan 2 = +63172 ' +099T2 = +  2830 tan 2 = ±0 ' tSa2 °
 = ±28?01 % = ±56?02 cos  = +0.8828
a  B ~ F  ° 8172 = n525
~ 7T^ 7~Z ~i ~ 2 X 0.9992 X 0.7793 u *
2 sm(co + 12) cos 2 K
Assembling the elements in the usual notation and the Thiele
Innes constants, we have:
108
THE BINARY STARS
P =
T =
a =
e =
« =
i =
S2 =
A =
B =
F =
G =
51.33 years
1925.68
0'/525
0.9102
87?35
±56?02
4?95
0.0011
+0.2939
0.5233
0.0317
CONSTRUCTION OF THE APPARENT ELLIPSE FROM THE
ELEMENTS
It is sometimes desirable to be able to construct the apparent
ellipse from the elements of the true orbit. This construction
is easily and quickly effected in the following manner:
Fig. 6. — The true and apparent orbits of a double star. (After See.)
Take the point (Fig. 6), at the intersection of two rectangular
axes, OX and OY, as the common center of the true and pro
jected orbits. Draw the line OQ making an angle equal to 12
with the line OX, counting from 0°. Lay off the angle <a from
the line Ofi, starting from the extremity 12 between 0° and 180°
THE ORBIT OF A VISUAL BINARY STAR 109
and proceeding in the direction of the companion's motion (clock
wise, that is, if the position angles decrease with the time,
counterclockwise, if they increase with the time). This will
give the direction of the line of apsides, AOP, in the true orbit.
Upon this line lay off OS, equal to ae, the product of the eccen
tricity and the semiaxis of the orbit, using any convenient
scale, and OP and OA, each equal to a. The point S lies between
and P, and P is to be taken in the quadrant given by applying
(a to $2 as described above. Having thus the major axis and the
eccentricity, the true ellipse is constructed in the usual manner.
Now divide the diameter yOO of this ellipse into any con
venient number of parts, making the points of division symmet
rical with respect to 0, and draw chords 6/36', etc., perpendicular
to the line of nodes. Measure the segments /36, 06', etc., and
multiply the results by cos i. The products will evidently be
the lengths of the corresponding segments /9&i, fib 2 , etc., in the
projected ellipse, and the curve drawn through the points &i,
C\, di, . . . will be the desired apparent orbit.
To find the position of the principal star in the apparent
ellipse draw through S a line perpendicular to the line of nodes,
and on it lay off the length ae sin a> cos i from the line of nodes.
This will give S', the point required. Lines through S' parallel to
OX and OY will be the rectangular axes to which position angles in
the apparent orbit are referred, and the position angle of the com
panion at any particular epoch may be obtained by laving off the
observed position angle. The line OS' extended to meet the ellipse
demies P', the projection of the point of periastron passage.
DIFFERENTIAL CORRECTIONS
If sufficient care is exercised in the construction of the apparent
ellipse, methods like those described will, as a rule, give a pre
liminary orbit that will satisfy the observed positions within
reasonable limits and that will approximate the real orbit
closely enough to serve as the basis for a least squares solution.
It should be emphasized that a satisfactory representation
of the observed positions does not necessarily imply a correct
orbit when the arc covered by the observations is comparatively
small. The percentage of error inherent in double star measures
is so great that, if the observed arc is less than 180°, it will
generally be possible to draw several very different ellipses
110 THE BINARY STARS
each of which will satisfy the data of observation about equally
well. In general, it is not worth while to compute the orbit of a
double star until the observed arc not only exceeds 180°, but also
defines both ends of the apparent ellipse.
It may not be amiss to add a few words on the practice, all
too common, of giving a fictitious appearance of accuracy to the
values of the prehminary elements by the use of unwarranted
decimals. To give the angular elements, the eccentricity, or
the semimajor axis to three or four decimals when even the first
decimal is uncertain, or the period and time of periastron passage
to the second or third decimal when the latter may be in doubt
by years and the former by decades, or in extreme cases, by
centuries, adds nothing to the real accuracy of the results and
does not inspire confidence in them.
Many computers are content with a preliminary orbit; but
it is advisable to correct these elements by the method of least
squares whenever the data are sufficient for an investigation
of the systematic errors of observation.
The position angle is a function of the six elements
QAf)°
ft, i, a?, e ( = sin <£), T and n = — =
and the required differential coefficients for the equations of
condition can be computed with all necessary accuracy from
the approximate formula
AAft + BAo> + CM + DA<p f FAM + GA M + (C  0) = (51)
where Aft, Aoj, etc., are the desired corrections to the elements,
and (C — 0) is the difference between the position angle com
puted from the preliminary elements and the observed angle
and A, B f etc., are the partial differential coefficients. The
eccentric angle <p, defined by sin <p = e, and the mean anomaly
M [= fi(t — T)] are substituted for the eccentricity (e) and the
time of periastron passage (T).
The corresponding equation for the distances may be written
hAa + bAco + cAi + dA<p + fAM + gAfi + (C  0) = (52)
the difference between the computed and observed distance
forming the absolute term.
The values of the partial differential coefficients are derived
from the equations for 6 and p (pages 79, 80). It has been the
THE ORBIT OF A VISUAL BINARY STAR 111
common practice to base the corrections to all the elements
except the semimajor axis, solely upon the residuals for position
angle, but it is sounder, theoretically at least, and in many cases
practically as well, to utilize the residuals both in angle and in
distance for the corrections to the five elements w, i, <p, M and n
by combining the two sets of equations of condition after elimi
nating A £2 from the first set and Aa from the second. If this
combination is made, the term (C — 0) in the equations for
position angle must be expressed in circular measure, which is
accomplished by multiplying the angle residuals by the factor
p/57.3.
Comstock has put the expressions for the differential coeffi
cients into the form given below. The residuals in angle are
assumed to be expressed in circular measure, and the expressions
are so formulated that the solution of the equations of condition
by the method of least squares will give the correction to the
semimajor axis in seconds of arc, and those to the other elements
in degrees.
Introduce the auxiliary quantities m, k and a defined by
m = p/57.3 = (8.2419)p,
k = (2 + sin cp cos v) sin E, and
a = — m tan * sin (0 — Q.) cos (0 — £2).
The differential coefficients for the terms given in the central
column are:
Angle Terms Distance
A = +m A£2
B = +wf   cos i Aa> b = + <r sin i
W
C = +<r At c = +<r tan (0 — Q)
D = +.B( \k A<p d = \b(\k — ml J cos <p cos v
F = —El] cos^j AM /= — b(  J 2 cos^ — ml  J sin^>sin.E
G =  F (jt  T ) An g = f(t  T)
Aa h = \ —
a
If the ThieleInnes constants A, B, F, G, have been computed,
the following method will be found convenient, particularly if
112 THE BINARY STARS
tables for X and Y are available.* The equations of condition
take the form
Ax = XAA + YAF + P x Ae + Q x nAT + R x An\
Ay = XAB + YAG + P y Ae + Q^nAT 7 + ^Aw/ { 6)
where P* = +A^ + F^
de de
p > = +*%+<€
V * " *dM ^dM
B x = {t  !T)Qs
i2 v = (« TOQv
From the formulas already given,
M = E — e sin E
X = cos E — e
F = Vl  e 2 sin M =  sin v J
e = sin <p, \/l — e 2 = cos ^
we have, by differentiation,
dX
— = OOlfl \ —
de I cos 2 ^>(cos 2 <p — X sin ^>)
(54)
*L = looi ZF
de ' cos 2 ^>(cos 2 ^ — X sin ^>)
^ = 0.017453 , j^ — ^ c
aAf cos ??(cos 2 <p — X sin ^)
•g = +0.017453 COS / (X + 8in y j
aiM cos 2 <p — X sin ^>)
the units being 0.01 for Ae and 1° for AM.
The question of algebraic sign is always an important one in
computing differential coefficients. For the present formulas
van den Bos gives the following rules :
dX/de is always negative,
* Such tables have been issued, in an Appendix to Un. Obs. Circ. 71. The
method described is given in Circs. 68 and 86.
THE ORBIT OF A VISUAL BINARY STAR 113
dY/de is positive when X and Y have the same sign, negative
when they have opposite signs,
dX/dM has the sign contrary to that of Y,
dY/dM is negative only when X is negative and numerically
greater than e.
These rules follow from the fact that cos <p and the expression
(cos 2 <p — X sin <p) — which is simply a transformation of the
expression 1 — e cos E derived by differentiating Kepler's
equation — are always positive, and, for the last two formulas,
from simple geometrical considerations. The sign of X is given
directly in the tables referred to; that of Y is always the same as
that of M , the mean anomaly.
The special advantage of the method arises from the fact that
the values of dX/de, etc., can be read off directly from the tables
of X and Y, for the value of the differential coefficient differs
from the mean of the preceding and following first difference,
in the tables (taken horizontally, for de and vertically for dM),
by only onesixth of the third difference, a negligible quantity.
If these tables are not available, the differential expressions
dX/de etc., may be computed from the formulas given above,
special attention being given to the units for Ae and AM and to
the algebraic signs.
SPECIAL CASES
All methods based upon the construction of the apparent
ellipse fail when the inclination of the orbit plane is (within
the limit of error of observation) 90°; for the apparent ellipse is
then reduced to a straight line and the observed motion is entirely
in the distance, the position angle remaining constant except for
the change of 180° after each occultation or apparent merging
of the two components into a single image. Such a Hmiting
case is actually presented by 42 Comae Berenices (21728).
Many other pairs are known in which the orbit inclination is
only slightly smaller.
In the hmiting case, the elements, except 12 which is obviously
given by the observed position angle, and i (90°), must be
derived entirely from the observed distances; in other cases
special methods may be devised which will vary with the pecul
iarities of the observed motion but which will depend in large
part upon the observed distance.
114
THE BINARY STARS
The first orbit for 42 Comae Berenices was published by Otto
Struve in 1875,* but his paper contains no statement as to the
methods used in obtaining his preliminary set of elements, and,
in 1918, 1 was unable to find in print any solution of the problem.
Simple graphical methods for finding the elements P, T, and e
from the curve of observed distances at once suggested them
selves; but methods for deriving the values of a and w were not
immediately apparent. In the first edition of this book I
outlined a method for obtaining them which Prof. F. R. Moulton
had kindly sent me in manuscript. More recently other methods
have been published by Prof. Kurt Laves, f Dr. F. C. Henroteaut
Fig. 7.
Apparent and true orbits and interpolating curve of observed distances
for a binary system in which the inclination is 90°.
and Prof. R. T. Crawford. § The methods by Laves and by
Crawford are the simplest ones, and the latter will be given here
in conjunction with the methods for deriving the elements
P, T, and e.
In Fig. 7 let the ellipse represent the true orbit and the line
T'C'T\, its projection upon the plane perpendicular to the line
of sight. Let the curve APA be the interpolating curve of
observed distances, obtained in the usual way by plotting the
distances against the times and drawing the most probable
smooth curve to represent the plotted points.
The revolution period may be read directly from this curve
and the accuracy of its determination will increase with the
number of observed revolutions.
In the true orbit, let S be the position of the primary star and
C, the center of the ellipse. Then the points C and S f on the
projected orbit are known for T'C {— Ti'C) must be half the
* Mon. Not. R.A.S., 35, 367, 1875.
t A.J. 37, 97, 1926.
X Handbuch der Astrophysik, 6, 338, 1928.
§ Lick Obs. BuU. 14, 6, 1928.
THE ORBIT OF A VISUAL BINARY STAR 115
amplitude of the curve of distances and S' must be the apparent
position of the primary star.
The points on the curve of distances which correspond to the
points P and Pi of the true orbit must be separated by exactly
half of the revolution period and their distances from the line
C'CE must be equal in length and of opposite sign. The point
corresponding to periastron, P, must lie on the same side of this
line as S' t and on the steeper branch of the curve as the rate
of change of distance is greater near periastron than near
apastron. In practice these two points are readily found by
cutting a rectangular slip of paper to a width equal to half that
of the period on the adopted scale and sliding it along the curve
until the edges, kept perpendicular to the line C'CE, cut equal
ordinates (with respect to C'CE) on the curve; or we may adopt
Laves's suggestion to "draw the central axis of the graph and
make a tracing of the time graph of distances, turn it through
180° about the central time axis and then advance the tracing
by half the period of the orbit. Of the four points of intersection
of graph and tracing it is easy to find that pair which is separated
by half the period." In either of these ways the positions of the
points P' and Pi' are obtained.
Since ratios are not altered in the projection we have
6 = CP' ^ 55 ^
To obtain the two remaining elements a and w, Crawford makes
use of a relation involving the perpendicular from the center
of the ellipse on a tangent to the ellipse. In the figure let CH
be such a perpendicular and call its length p. Let co (which is
here our element u) be the angle the tangent line makes with the
minor axis. Then, if a and 6 are the major and minor semiaxes,
respectively, from the properties of the conic it can be shown that
p2 _ 2 cog 2 w _j_ ^2 gm 2 w
which can be written
p2 = a 2 — (a 2  6 2 ) sin 2 w = a 2  a 2 e 2 sin 2 «. (56)
But, from the figure,
CK = m = CS cos ct) = ae cos cu
116 THE BINARY STARS
Hence
and
cos u> — — (57)
t tft
sin2a, = 1 ^ < 58 >
Substituting (58) into (56) and solving for o, we have
«>£
=^ (59)
p is C'T', and m is C'S', both of which are known.
The order of solution is, then, Eqs. (55), (59), and (57) which
give e, a, and «, respectively.
If the major axis is directed toward the Earth, the points
C", P', and S' coincide and « = 90°. Equation (55) then becomes
indeterminate, but at the same time p = b and m = 0. Hence,
from Eq. (59) we have
«  Jr^p (60)
To determine e, let t be the ratio of the intervals of time
from elongation to elongation such that t < 1. Then from
the law of areas we have t equal to the ratio of the area described
by the radius vector in the true orbit in the shorter interval to the
area described by the radius vector in the longer interval, or
^7T06
— abe
+
e
}4,irab + abe
e
e =
IT 1 "
2 1 
— T
f T
whence
ir 1 —  T
(61)
The time of periastron passage, T, is the middle instant of the
shorter interval between two consecutive elongation times.
If the intervals are all equal, r — 1, e = 0, and the orbit is a
circle, in which case the elements o> and T lose significance.
When a preliminary set of elements has been derived, improved
values may be computed by the method of least squares, using
THE ORBIT OF A VISUAL BINARY STAR 117
Comstock's expressions for the differential coefficients in the
equation for distance.
THE ORBIT OF e EQUULEI
An excellent example of an orbit whose computation was
made possible by taking advantage of the special features of
the observed motion is that of e Equulei, published by Russell
in 1917.* The apparent orbit of this system is an extremely
narrow and elongated ellipse. Fortunately the double star was
discovered by Struve in 1835, when the angular separation
was only 0"35. In later years the companion moved out to a
maximum elongation of 1"05 and then in again until, in 1917,
the pair could not be resolved by any existing telescope. Plotting
the distances (using mean places) against the times, Russell
noted that the curve was practically symmetrical with respect
to the maximum separation point. It follows that the line
of apsides in the true orbit must be approximately coincident
with the line of nodes, or that w = 0. Further, the mean of
the position angles for a few years on either side of the time of
elongation gives a preliminary value for the angle ft, and the
elongation time itself is the epoch of apastron passage, which
may be taken in place of the epoch of periastron as one of the
orbit elements. It is also apparent that the inclination of the
orbit is very high and a preliminary value for this element may
be assumed. This leaves the three elements, a, e, and P, which
Russell finds as follows: Let
yx = the maximum elongation distance
T' = the corresponding epoch (i.e., apastron)
y" = the distance at any other time t"
E = the corresponding eccentric anomaly in the true orbit.
Then we have
a(l + e) = yi f a(cos E — e) = — y"
M = E  e cos E, and (t"  T) = 180°  M
which determine a and P in terms of e.
Assume values of e and compute a and P, repeating the
process until those values result which represent the curve of
the observed distances.
The preliminary elements obtained by these processes Russell
corrected differentially, a, e, T and /* from the observed distances,
* A. J. 30, 123, 1917.
118 THE BINARY STARS
i and Q. from the observed angles; to (= 0) being assumed as
definitely known,*
SYSTEMS IN WHICH ONE COMPONENT IS INVISIBLE
Luminosity, Bessel said long ago, is not a necessary attribute
of stellar mass, and it may happen that one component of a
double star system is so feebly luminous as not to be visible in
existing telescopes. If the orbit is one of short period and the
inclination of its plane sufficiently high, the system may be
detected by the spectroscope, by the methods to be discussed
in the following chapter. In other instances the companion's
presence may be revealed by a periodic variation in the bright
star's proper motion, the path described by it upon the celestial
sphere becoming a cycloid instead of the arc of a great circle.
A system of the latter type is most readily detected when the
proper motion is large, and it is of course essential that the
motion be accurately determined.
Variable proper motion was actually recognized in the stars
Sirius and Procyon, nearly a century ago, and was explained by
Bessel as the effect of the attraction of such invisible com
panions. Orbits, referring the motion of the bright star to the
center of gravity of a binary system, were thereupon computed
for these stars by C. A. F. Peters and A. Auwers. Bessel's
hypothesis was proven to be correct by the subsequent dis
covery of a faint companion to Sirius by Alvan G. Clark (in
1861), and of a still fainter companion to Procyon by Schaeberle
(in 1896). The relative orbit of the companion to Sirius has
been computed from the micrometer measures, and the elements
are consistent with those determined from the proper motion of
the bright star. There is no question but that this will also prove
to be the case in the system of Procyon when the micrometer
measures permit an independent determination of its orbit.
Dark companions to fi. Orionis, a Hydrae, and a Virginia
have also been suspected from supposed irregularities in the
proper motions, but closer examination of the data has not
verified the suspicion. Since cases of this kind will probably
always be very exceptional, the formulas for their investiga
* More recent observations indicate the need of revision of Russell's
elements.
THE ORBIT OF A VISUAL BINARY STAR 119
tion will not be considered here. Those who are interested in
their development are referred to the original memoirs.*
The presence of invisible companions in several wellknown
double star systems has also been revealed by the observed
periodic variations in the motion of one of the visible com
ponents. In one of these, e Hydrae, the primary star was
later found to be a very close pair whose components complete
a revolution in about fifteen years, and Seeligerf has shown
that the orbital motion in this close pair fully accounts for the
irregularities observed in the motion of the more distant com
panion. Another of these systems, £" Cancri, consists of three
bright stars, two of which revolve about a common center in a
period of approximately 60 years, while the third star revolves
with this binary system in a much larger orbit. Seeliger has
shown that the irregularities observed in the apparent motion
of this third star may be explained on the hypothesis that it is
accompanied by an invisible star, the two revolving about a
common center in circular orbits with a period of 18 years.
The system then, would, be a quadruple one.
Again, Norlund, in the course of his investigation of the orbit of
£ Ursae Majoris, in 1905, discovered a perturbation of 1.8 years
period with an amplitude of 0"05. He attributed this to the
presence of an invisible companion to the brighter component,
the two bodies revolving in an orbit inclined nearly 90° to the
plane of projection. Such a companion had, as a matter of
fact, been discovered by Wright, five years earlier, from the vari
able radial velocity of the bright star, but Norlund was unaware
of the discovery when he announced his result. Van den BosJ
has made a complete investigation of this triple system, § utilizing
* Bessel, A. N. 22, 145, 169, 185, 1845.
Peters, A. N. 32, 1, 17, 33, 49, 1851.
Auwers, A. N. 63, 273, 1865 and Untersuchungen uber verdnderliche
Eigenbewegung, 1. Theil, Konigsberg, 1862; 2. Theil, Leipzig, 1868. See
also A. N. 129, 185, 1892.
t A. N. 173, 321, 1906.
t Mem. de I'Acad. Roy. des Sciences et des Lettres de Danemark. Sec. des
Sci. 8 me Serie, 12, No. 2, 1928.
§ It is really a quadruple system, for an invisible companion to the fainter
visual component was discovered in 1918 from spectrograms taken at the
Lick Observatory. Berman's orbit (Lick Obs. Bull. 15, 109, 1931), however,
shows that the revolution period of this pair is only 3.98 days. No sensible
perturbation in the visual orbit can be produced by it.
120 THE BINARY STARS
both the micrometric and photographic measures of the bright
pair and the spectrographs observations of the primary star
and its invisible companion. There are irregularities in the
observed motion of 70 Ophiuchi which have been regarded as
due to the perturbations produced by a third body, but a really
satisfactory solution of the orbit has not yet been published.
Finally, we may refer to Comstock's investigation* of the orbital
motion in the system f Herculis from which he concludes that
small irregularities in the areal velocity of the bright pair may be
represented as the effect of an invisible companion to one com
ponent, having a periodic time of 18 years and an amplitude less
than 0"1. Comstock, however, points out that when the sys
tematic errors of the observers are determined and allowed for,
the orbit, without the assumption of a third body, "satisfies the
observations within the limits of error commonly deemed satis
factory." The paper is an excellent example of the method by
which systematic errors should be investigated in the computa
tion of a definitive double star orbit.
It is probable that the invisible companion in such a system
as that of f Herculis revolves, like the bright components, in
an elliptic, rather than a circular orbit; and it is not at all
improbable that the plane of this orbit is inclined at a greater
or less angle to the plane of the orbit of the visible system.
To determine the eccentricity and the inclination, however,
would greatly complicate the problem and the precision of the
observational data is not sufficient to warrant such refinements
unless, as in the case of £ Ursae Majoris, spectrographs observa
tions are also available. In practice, it has been found satis
factory, in general, to assume that the invisible body moves in a
circle in the plane of the orbit of the visible stars of the system.
This assumption leaves but two elements to be determined, the
period and the radius or semiamplitude, and the formulas for
these are quite simple. Comstock's formulas for the companion
in the system of f Herculis, for example, are as follows :
Let 6, p, represent the polar coordinates of the visible com
panion referred to the primary star; \f/, r the corresponding
coordinates of the center of gravity of the assumed system
(i.e., the system comprised of the secondary bright star and
its dark companion) referred to the same origin; and v, a, the
coordinates of the visible companion referred to the center of
* A. J. 30, 139, 1917.
THE ORBIT OF A VISUAL BINARY STAR 121
mass of itself and its dark companion. Then we shall have
from the geometrical relations involved,
p 2 = r 2 _j_ a 2 __ 2ar cos (v  f)
S = ^ +  sin {v  $) (62)
P
If we assume that a/r and r/p are quantities whose squares
are negligibly small, we have by differentiation
A6 J4> , .Jdv d\f\ . , dr
Since the assumed system is circular, a and dv/dt are constant
d\l/
quantities, r 2 rr is also a constant, and a is so small that, in
the second member of the equation, we may write for ^ and
p in place of r without sensible error. If, further, for brevity,
we put d4//dt = K/p 2 and k = dv/dt, the equation takes the
form
p 2 ^ = K + a(k P + J cos(w  0)  a sin(y  0)^? (64)
RECTILINEAR MOTION
The relative motion in some double stars is apparently recti
linear and it is desirable to have criteria which shall enable us
to decide whether this results from the fact that the orbit is a
very elongated ellipse, or from the fact that the two stars are
unrelated and are changing their relative positions by reason
of the difference in their proper motions. One excellent test,
which has been applied by Lewis to many of the Struve stars,
is to plot the path of the companion relative to the primary
and note whether its motion along that path is uniform or whether
it becomes more rapid as the distance between it and the primary
diminishes. In the former case the two stars are independent,
in the latter, they are physically related.
A more rigorous test is the one applied, for example, by
Schlesinger and Alter* to the motion of 61 Cygni. If the motion
is uniform and in a straight line, the position angles and distances
of the companion referred to the primary may be represented
by the equations
*Publ. Allegheny Obs. 2, 13, 1910.
122
THE BINARY STARS
p 2 = 2 __ (t _ y)2 m !
m,
tan (0  0) = £(<  7)
(65)
in which a is the perpendicular distance from the primary, con
sidered as fixed, to the path of the companion; <f> is the position
angle of this perpendicular; T, the time when the com
panion was at the foot of the perpendicular, and m, the
annual relative rectilinear motion of the companion. Approxi
mate values for these four quantities may be obtained from
a plot of the observations and residuals may then be formed
Fig. 8. — Rectilinear motion.
by comparing the positions computed from the formulas with
the observations. If these residuals exhibit no systematic
character, rectilinear motion may be assumed; if they show a
systematic course a closer examination is in order to decide
whether this is due to chance or to orbital motion. In the
latter case, the indicated curve must be concave to the primary
and the systematic run of the residuals should be quite uni
form. In any event, a least squares solution may be made to
obtain more precise values for the quantities a, <£, T and m.
For this purpose, differentiate Eqs. (65) and introduce the
values sin (0  </>) = ^^ — } , cos (0 <*>) =  p (see Fig. 8) ; we
THE ORBIT OF A VISUAL BINARY STAR 123
thus obtain the equations of condition in the form given by
Schlesinger and Alter:
— cos (6 — <j> )Aa — sin (0 — <f> )(t — To) Am
+ sin (0 — <t>o)moAT + Ap = v p
+ sin (0 — <£ )Aa — cos (0 — <j> )(t — T )Am
+ cos (0 — 0o)w o A7 T — pA<f> + pA0 = v B \
(66)
in which the subscript o indicates the preliminary values of the
elements, Ap and Ad the deviations from the approximate
straight line, and v p and v e the residuals from the definitive
values of the elements.
References
In addition to the papers cited in the footnotes to the chapter, the student
of double star orbit methods will find the following of interest:
Klinkerfues: Uber die Berechnung der Bahnen der Doppelsterne, A.N.
42, 81, 1855.
: Allgemeine Methode zur Berechnung von Doppelsternbahnen,
A.N., 47, 353, 1858.
Thiele: tJber einen geometrischen Satz zur Berechnung von Doppel
sternbahnen, u. s. w., A.N., 52, 39, 1860.
: Unders0gelse af Oml0bsbevaegelsen i Dobbelstjernesystemet
7 Virginis, Kj0benhavn, 1866.
Neue Methode zur Berechnung von Doppelsternbahnen., A.N., 104,
245, 1883.
Seeligeb: Untersuchungen iiber die Bewegungsverhaltnisse in dem drei
fachen Sternsystem f Cancri, Wien, 1881.
: Fortgesetzte Untersuchungen liber das mehrfache Sternsystem f
Cancri, Miinchen, 1888.
Schorr: Untersuchungen iiber die Bewegungsverhaltnisse in dem drei
fachen Sternsystem £ Scorpii, Miinchen, 1889.
Schwarzschild: Methode zur Bahnbestimmung der Doppelsterne, A.N.,
124, 215, 1890.
Rambaut: On a Geometrical Method of Finding the Most Probable Appar
ent Orbit of a Double Star, Proc. Roy. Dublin Society, 7, 95, 1891.
Howard: A Graphical Method for Determining the Apparent Orbits of
Binary Stars, Astronomy and Astrophysics, 13, 425, 1894.
Hall: The Orbits of Double Stars, A.J., 14, 89, 1895.
See: Evolution of the Stellar Systems, Vol. 1, 1896.
Leuschner: On the Universality of the Law of Gravitation, University of
California Chronicle, 18, No. 2, 1916.
Andre: Traite d'Astronomie Stellaire, Vol. 2.
Also the chapters on double star orbits in such works as Klinkerfues
Buchholz, Theoretische Astronomie; Bauschinger, Die Bahnbestimmung
der Himmelskorper; Crossley, Gledhill, and Wilson, A Handbook of
Double Stars.
124 THE BINARY STARS
Dobebck, W.: On the Orbit of £ Bootis, A.N. 214, 89, 1921.
Comstock, G. C. : On the Determination of Double Star Orbits from Incom
plete Data, A.J., 33, 139, 163, 1921.
Meyermann, B.: Eine neue graphische und eine halbgraphische Methode
zur Bestimmung von Doppelsternbahnen, A.N. 215, 179, 1922.
: Zur Bestimmung von Doppelsternbahnen, A.N. 228, 49, 1926.
Dawson, B. H.: Provisional Elements of the Binary Star h50H with a Note
on the Method Employed, A.J. 36, 181, 1926.
Nassau, J. J., and P. D. Wilkins: Graphical Determinations of Orbits of
Visual Binary Stars, A.J. 38, 56, 1928.
Henroteau, F. C. : Double and Multiple Stars, Handbuch der Astrophysik,
Bd. VI, 2 ter Teil, Chapter 4, 1928.
Parvulesco, C: M6thode nouvelle pour calculer des Orbites d'Etoiles
doubles, Bull. Lyon Obs. 10, 49, 1928.
: Contribution a la determination de l'orbite apparente d'une Etoile
double, d'apres la variation de Tangle de position, Bull. Lyon Obs. 12,
122, 1930.
Vahlen, Th.: Doppelsternbahn aus sieben Beobachtungen, A.N. 233, 217,
1928.
Kerrich, J. E. : A Method for the Computation of the Orbital Elements for
Certain Binary Stars, Union Obs. Circ. 82, 123, 1930.
Volet, Ch.: Application de la methode des moindres carrees au calcul des
orbites d'&oiles doubles, C.R. 192, 482, 1931.
: Methode pour le Calcul des Orbites d'Etoiles Doubles Visuelles,
Application a l'Orbite du Compagnon de Sirius, Butt. Astron., Ser. 2,
7, 13, 1931.
Russell, H. N.: A Rapid Method for Determining Visual Binary Orbits,
Mon. Not. R.A.S. 93, 599, 1933.
De Sitter, W. : On the Solution of Normal Equations, Ann. Cape Obs. 12,
Pt. 1. 160173.
CHAPTER V
THE RADIAL VELOCITY OF A STAR
By J. H. Moore
The observations treated in the preceding chapters concern
only that part of the star's actual motion in space, which appears
as change of position in a plane perpendicular to the line joining
the observer and star. Of the component directed along the
line of sight, called the star's radial motion, the telescope alone
gives no indication. In fact, the possibility of detecting radial
motion was recognized less than a century ago, and the methods
of its measurement belong distinctly to another and newer
branch of astronomy, known as astrophysics. Moreover,
observations of the rate of change of position of a star on the
celestial sphere can be translated into linear units, such as
kilometers per second, only if the star's parallax is known, while
measures of radial velocity by the method to be described,
are expressed directly in kilometers per second and are inde
pendent of the star's distance.
The determination of the radial velocity of a light source,
such as a star, is made possible by two wellknown properties
of light; viz., that it is propagated as a wave motion, and with
a definite and finite velocity. We are not concerned with
the properties of the hypothetical medium, called the ether,
in which these waves move, nor with the nature of the dis
turbance in the ether, whether it be mechanical or electromag
netic. For our purpose it is sufficient to know that in this
medium, or in interstellar space, the velocity of light is about
299,796 km/sec. and that the wellknown laws of wave motion
hold for light waves.
In 1842, Christian Doppler called attention to an effect
upon the apparent length of a wave which should result from
a relative motion of the source of the waves and the observer.
This result was independently reached and further developed,
especially with reference to light waves, some six years later
by the great French physicist, Fizeau. According to the
125
126 THE BINARY STARS
DopplerFizeau principle, when the relative motion of the light
source and the observer is such, that the distance between the
two is increasing or decreasing, the length of the waves received
by the observer will be longer or shorter, respectively, than the
normal length of these waves.
It is readily shown that the change in wave length is directly
proportional to the normal length of the wave and to the ratio
of the relative velocity of source and observer to the velocity
of propagation of the waves. Moreover, for light waves the
change is the same whether the source, or observer, or both are
moving and depends only upon the relative velocity of the two.
Let us denote by v the relative radial velocity in kilometers
per second of a star and observer, where v is considered positive
when the distance between the two is increasing and negative
when this distance is decreasing. Call X' the wave length of a
monochromatic ray reaching the observer, whose normal wave
length, as emitted by the star, is X.
Then from the DopplerFizeau principle, X 7 — X:X: :v: 299,796;
or X' — X = Xv/299,796 (if v is +, X' is greater than X) ; or, writing
AX for the change in wave length (X' — X), we have for the
relative radial velocity of star and observer
299,796AX „.„
v = ^ (1)*
The determination of the radial velocity of a star rests then
upon a knowledge of the velocity of light and of the wave
lengths of certain definite rays emitted by a source at rest, and
the measurement of the apparent wave lengths of those same
rays received from a star. In short, the problem reduces to one
of measuring AX with the greatest possible precision. For this
purpose the micrometer, with which we have become familiar,
is replaced by the spectroscope. This wonderful instrument
originating in the physical laboratory has developed a whole
new science, spectroscopy, with an extensive and technical
literature of its own. In this chapter we shall only call atten
tion to some of the elementary principles of spectroscopic
analysis and give a very brief survey of the spectrographic method
* It may be shown from the theory of relativity that this formula holds
for the relative velocity of source and observer where this is small in com
parison with the velocity of light. This condition is fulfilled by the stellar
light sources with which we are here concerned.
THE RADIAL VELOCITY OF A STAR 127
as applied to the determination of stellar radial velocities.
The student who wishes to pursue the subject further will
find a list of references to extended treatment of the various
topics at the end of this chapter.
Since stellar light sources are very faint in comparison with
those available in the laboratory, it is necessary to employ for
this special problem the spectroscope which is the least wasteful
of light. For this reason the prism spectroscope is the only
one of the various laboratory forms which is at present generally
applicable to stellar spectroscopy* and therefore we limit our
discussion to this particular type.
The essential parts of a laboratory spectroscope and their
principal functions are briefly as follows: Light from the
source to be studied is brought to a focus by a condensing lens
on the narrow slit of the spectroscope. After passing through
the slit, the rays are rendered parallel by an achromatic con
verging lens, called the collimator lens. The rays then strike
a glass prism, placed with its apex parallel to the length of
the slit, by which they are bent from their original direction.
It is here that we obtain the separation of the rays, since the
amount by which each ray is deviated by the prism is a func
tion of its wave length. The direction of the long red waves
is changed the least, while the shorter violet ones suffer the
greatest deviation. After each set of rays is collected and
brought to its corresponding focus by a second achromatic
converging lens, we shall have an orderly array of images of
the slit, each image formed by light of a definite wave length.
Such a series of images is called a spectrum of the source. The
spectrum may be viewed with an ordinary eyepiece, or the
second lens may be used as a camera lens, and the spectrum
be recorded on a photographic plate placed in its focal plane.
When the spectroscope is employed photographically, as it is
in practically all stellar work, it is called a spectrograph, and the
photograph obtained with it is a spectrogram.
If the slit is made extremely narrow, there will be very little
overlapping of the images and the spectrum is then said to be
* In recent years, grating spectrographs of high dispersion have been
successfully employed with large telescopes for studies of the spectra of the
brighter stars. This instrument has also proved more efficient for the
investigation of the spectra of the red stars on account of the greater dis
persion given by the grating in the region of longer wave length.
128 THE BINARY STARS
pure. It can be shown that the purest spectrum is obtained
when the incident rays fall upon the prism at such an angle
that they will be least deviated from their original direction
by the prism. It is well known that this position of minimum
deviation is also the one of maximum light transmission by the
prism; and it has the further advantage that any accidental
displacement of the prism produces the minimum displace
ment of the spectrum line. The prism or prisms of stellar
spectrographs are therefore always set at the angle of mini
mum deviation for the approximate center of the region of
spectrum to be studied.
Attention was called in an earlier chapter to two factors
which define the optical efficiency of a telescope for the sepa
ration of close double stars, viz.: (a) the resolving power of
the objective, (6) the magnification or linear distance between
the two images at the focus of the objective. These same
factors form a convenient basis for the comparison of the resolving
powers of two spectrographs. Here, however, we are con
cerned with the separation of two images of the slit formed
by light of different wave lengths. The resolving power of
a spectrograph is, therefore, denned as the minimum difference
of wave length between two lines for which the lines will just be
separated. It is a function of the width of slit, the wave length,
and the difference between the maximum and minimum lengths
of path of the rays in the prism. The magnification, called
the linear dispersion of the spectrograph, is expressed, as the
number of wave length units per unit length of spectrum and
depends upon the wave length of the ray, the optical constants
for the prism system and the focal length of the camera lens.
When the slit of a spectroscope is illuminated by the light
from an incandescent solid, such as the filament of an incan
descent lamp, or from an incandescent gas under high pressure,
the spectrum consists of an unbroken band of color; that is, a
continuous spectrum. An incandescent gas or vapor under
low pressure gives a spectrum consisting of isolated bright
line images of the slit, a bright line spectrum, the bright lines
indicating that radiations of certain definite wave length are
emitted by the gas. Each chemical element, in the gaseous
state, when rendered luminous in the electric arc, electric spark,
flame, or vacuum tube, gives its own set of bright lines which
are characteristic of this element alone and whose wave lengths
THE RADIAL VELOCITY OF A STAR 129
remain constant for a source at rest under the same conditions
of temperature, pressure, etc.
An incandescent gas has the property not only of radiating
light of certain definite wave lengths, but also of absorbing,
from white light passing through it, the rays of precisely those
same wave lengths. If the temperature of the incandescent
gas is lower than that of the source behind it, the continuous
spectrum will be crossed by relatively dark lines whose posi
tions agree exactly with the bright line spectrum characteristic
of the gas. This relation existing between the emission and
absorption of a gas is known as Kirchhoff's law, and the type
of spectrum described is termed an absorption spectrum.
The three principles just stated obviously lead to a simple
and direct method of analyzing the chemical constituents of
gaseous light sources and of furnishing information as to their
physical conditions. Nebulae of a certain class, for example,
give bright line spectra, indicating that they are masses of
luminous and extremely rarefied gases. Most of the stars,
including our own Sun, give absorption spectra, showing that
the light emitted by a central glowing core has passed through
a surrounding atmosphere of cooler vapors. The presence in
the atmospheres of the Sun and stars of most of the known
chemical elements has been recognized from the lines in the
spectra of these objects. In addition, there occur in these
spectra many lines, which have not yet been identified with
those of any known element.
The length of the light wave for each line is such a minute
fraction of a millimeter that spectroscopists have adopted as
the unit of wave length, the Angstrom, equal to 0.0000001 mm.
for which A is the symbol. Thus the wave length of the hydrogen
radiation in the violet is 0.0004340 mm. or 4340A.
Measures of the wave lengths of the lines in a star's spec
trum secured with the prism spectrograph, are readily effected
by a comparison of the positions of the stellar lines with those
from a source the wave lengths of whose lines are known. To
accomplish this the light from a suitable source (for example,
the iron arc) is made to pass over very nearly the same path
in the spectrograph as that over which the star's light travels,
and the spectrum of this source, termed the comparison spectrum,
is recorded on each side of the star spectrum.
130 THE BINARY STARS
When the spectra of a number of stars are examined, it is
found that they exhibit a great variety in the number and
character of their lines. From an examination of several
hundred stars by means of a visual spectroscope, Secchi, about
18661867, was able to arrange their spectra under four types.
While exhibiting very well the most prominent characteristics
of stellar spectra, his system is insufficient for portraying the
finer gradations, which the photographic method has brought
to light. The classification now in general use among astro
physicists was formulated by Prof. Pickering, Miss Maury,
and Miss Cannon from the very extensive photographic survey
of stellar spectra made at the Harvard College Observatory and
at the Harvard station at Arequipa, Peru. It is based upon the
observed fact that certain groups of lines have a common behav
ior. They make their appearance and increase or decrease in
intensity at the same time, so that a more or less orderly sequence
of development from one type of spectrum to another is indicated.
A condensed outline of this system will serve to indicate its
chief features. The main divisions are represented by the
letters, O, B, A, F, G, K, M (R, N, S). Classes B to M, in the
order given, form a continuous sequence, and types intermediate
between the main ones are indicated by numbers 1 to 9 inclusive.
Class O undoubtedly precedes Class B, but its subdivisions are
still provisional and are indicated by small letters a to e. Classes
R and N appear to form a branch from the main sequence begin
ning at Class K, while Class S seems to be still another such
offshoot.
In Class O, subdivisions Oa to Oe show faint continuous spectra
upon which are superposed bright bands. The lines of hydrogen
and helium are bright in the beginning of the class but dark in the
later subdivisions. Characteristic lines of the class are those of
ionized helium,* and doubly and trebly ionized oxygen, nitrogen,
* On the basis of the atomic model that considers the atom to consist of a
positively charged nucleus about which revolve negatively charged units
called electrons, it is possible to picture the manner in which the atoms
radiate energy of definite frequencies corresponding to the different spectral
lines. Each atom in the neutral state has a definite number of electrons —
hydrogen 1, helium 2, lithium 3, etc. — and to the atom of each element in
this state there corresponds a characteristic spectrum. If one electron is
removed from the atom, the latter is said to be singly ionized, if two are
removed, doubly ionized. Thus the atoms of ionized helium and doubly
ionized lithium have only one electron and give entirely different spectra
from those given by their neutral atoms. In the laboratory it is found
THE RADIAL VELOCITY OF A STAR 131
and silicon. Prominent features of Class B are the dark lines of
hydrogen and neutral helium. Near the end of the class the
helium lines weaken and they are absent in AO, while the lines
given by the ionized atoms of the metals magnesium, calcium,
iron, etc., begin to appear. The hydrogen lines reach their
maximum intensity early in Class A and steadily decrease in
strength through the remainder of the spectral sequence. The
lines of ionized calcium, H and K, and those of the metals
increase in prominence through this class and the subdivisions
of Class F. In Class G, which includes stars whose spectra
closely resemble that of the Sun, the H and K lines and the
numerous metallic lines are conspicuous features, whereas
hydrogen has become less prominent. Class K spectra show a
further weakening of the enhanced lines of the metals and
strengthening of the arc lines, especially those which appear in
the laboratory at lower temperatures. In Class M these low
temperature lines are still further strengthened as the hightem
perature lines decrease in intensity. This class is characterized
by the absorption bands of titanium oxide which first make
their appearance at K5 and increase in intensity through the
subdivisions of Class M. Stars of Class Me show, in addition,
bright hydrogen lines. To classes R and N belong stars whose
spectra of metallic lines are similar to those of M but which are
particularly characterized by bands of carbon and cyanogen.
Spectra of Class S are likewise similar to M in the strengthening
of the low temperature lines of the metals, but in this class the
bands of zirconium oxide are present in addition to those of
titanium oxide.
Stars of classes O and B are bluish white in color; those of
Class A, white; of F and G, yellow; of K, orange; of M, R, and S,
red; while the N stars are a deep red.
In Plate III are reproduced four stellar spectrograms secured
with the threeprism spectrograph of the D. 0. Mills Expedi
tion, at Santiago, Chile, which illustrate the different appear
ance of the spectra in the blueviolet region of classes B8, F,
G, and K5. On all of the spectrograms the bright line spectrum
that the lines from the ionized atom are stronger in the spark (enhanced)
as compared with their intensities in the electric arc, while in the latter the
lines of the neutral atom are stronger. The lines of the ionized atom are
therefore frequently referred to as enhanced and those of the neutral atom
as arc lines.
132
THE BINARY STARS
of the iron arc was photographed above and below the star
spectrum. The spectrum of v* Eridani (Figure a) of Class B8,
shows only the hydrogen line H7 (4340. 4 77 A) and the magnesium
i
si
"1 IHBIB
line (4481. 228 A), as the very faint metallic lines, some of which
appear on the original negative, are lost in the process of reproduc
tion. This star is a spectroscopic binary, and the spectra of
both stars are visible, so that each of the two lines mentioned
THE RADIAL VELOCITY OF A STAR 133
above is double. The strengthening of the metallic lines and the
decrease in intensity of H7 are shown in the spectrum of a Carinae
of Class F (Fig. b), while in the solar spectrum (Fig. c), of Class G,
and in that of a 2 Centauri (Fig. d), of Class K5, a further decrease
in Hy, the disappearance of 4481 A and a considerable increase
in the number and strength of the absorption lines of other
elements are noticeable.
The four spectrograms illustrate also the displacement of
the lines in star spectra as effects of motion in the line of sight.
The iron lines in the solar spectrum are practically coincident
with the corresponding lines of the iron arc, since the relative
radial velocity of the Sun and the observer is very small. The
iron lines in the spectrum of a Carinae are clearly displaced
from their normal positions, as given by the lines of the com
parison spectrum. This displacement is toward the red end
of the spectrum, and corresponds, therefore, to an increase in
the wave lengths of the star lines. Interpreted on the Doppler
Fizeau principle, this change is produced by a recession of the
star with respect to the Earth at the rate of +18.0 km/sec.
In the case of a 2 Centauri, the displacement of the lines is
toward the violet and corresponds to a velocity of approach of
—35.0 km/sec. As an example of the DopplerFizeau effect,
the spectrogram of the spectroscopic binary u 4 Eridani, is perhaps
the most striking. The two stars revolve about their common
center of mass in a period of 5.01 days, as shown by an extended
series of plates similar to this one. Due to their orbital motion,
the velocity of each star in the line of sight is continually chang
ing, giving rise to a continuous variation in the separation of the
lines of the two spectra. The spectrogram reproduced here was
taken at the time of maximum velocity of approach of one, and
the corresponding velocity of recession of the other component.
It shows, therefore, the maximum separation of the lines of
the two spectra. The relative radial velocity of the two stars
was 132 km/sec. Obviously, the lines of the two spectra will be
coincident when the motion of the two components is across the
line of sight, which occurs at intervals of 2.5 days.
It is well known that the wave lengths of spectral lines are
affected by other causes than that arising from radial motion
of the source. For example, it is found that an increase in
pressure of the emitting or absorbing vapor will in general shift
the lines toward the red. This effect, even with considerable
134 THE BINARY STARS
pressures, is small and is moreover not the same for all lines.
Of the many conditions which displace spectrum lines, radial
motion is the only one of which measures of stellar spectra
have furnished reliable evidence. *
Displacements of the stellar lines, with reference to those of
the comparison spectrum, may arise wholly or in part from
causes which are purely instrumental. Thus, if the starlight
and the artificial light do not pass over equivalent paths in
the spectrograph, or if a change in the relative positions of the
parts of the instrument occurs between the times of photograph
ing the stellar and the reference spectra, a relative displacement
of the lines of the two spectra will result. The firstnamed
source of error is an optical condition, to be met for all spectro
scopic measures, that is easily satisfied. With the conditions
of a fixed mounting and approximately constant temperature,
under which the spectrograph is used in the laboratory, the
second source of error need not be considered. When, however,
the spectrograph is applied to stellar observation, it is necessary,
in order to avoid undue loss of light, to mount it on a moving
telescope, and hence to subject the instrument to the varying
component of gravity and the changing temperature of a well
ventilated dome. The spectrograph must be so designed and
constructed that it will be free from appreciable differential
flexure in any two positions of the telescope, and provision must
be made against the disturbing effects of temperature changes
in the prisms and the metal parts of the instrument. Further,
in addition to the obvious requirement that the prisms and lenses
shall give good definition, they must be so chosen and arranged
as to give satisfactory resolving power with efficiency in light
transmission.
The earlier determinations of stellar radial velocities were
made entirely by the visual method. Although made by such
skilled observers as Huggins, Vogel, and others, the errors of
observation, except for a very few of the brightest stars, often
exceeded the quantities to be measured. After the introduc
tion of the photographic method of studying stellar spectra,
Vogel and Scheiner, at Potsdam, and later Belopolsky, at
Pulkowa, were able to measure the radial velocities of the
* A possible exception is the lengthening of the light waves in a strong
gravitational field, evidence of which has been found in the spectrum of the
companion of Sirius.
THE RADIAL VELOCITY OF A STAR 135
brightest stars with an average probable error of ±2.6 km/sec.
In 18951896 the problem was attacked by Campbell, who
employed a specially designed stellar spectrograph — the Mills
Spectrograph — in conjunction with the 36in. refractor of
the Lick Observatory. For the brighter stars, the probable
error of his measures was about ±0.5 km and for bright stars
whose spectra contain the best lines, the probable error was
reduced to ±0.25 km. Many improvements in stellar spectro
graphs have, of course, been made in the succeeding years,
but the standard of precision, set by his measures nearly 40 years
ago, represents that attained today for the same stars. The
advances which have been made in this time relate more to the
increased accuracy of the results for fainter stars.
Now this remarkable advance in the precision of the meas
ures made by Campbell was due not to the use of a great tele
scope but to the fact that his spectrograph was designed in
accordance with the important requirements mentioned in the
foregoing — excellence of definition and maximum light trans
mission, rigidity, and temperature control of the spectrograph —
and to improved methods of measuring and reducing the
spectrograms.
In order to understand more clearly the manner in which
the optical and mechanical requirements are met in practice,
a detailed description will be given of a modern spectrograph
which was designed to have maximum efficiency for the par
ticular problem of determining stellar radial velocities. A
view of the new Mills spectrograph attached to the 36in.
refractor of the Lick Observatory is presented in Plate IV.
The essential parts of this instrument are. the same as those
described for the simple laboratory spectrograph; namely,
the slit, collimator lens, prism, and camera lens, except that
here three 60° prisms of flint glass are employed. The prisms,
set at minimum deviation for 4500A, produce a deviation
of this ray of 176°. A rectangular box constructed of sawsteel
plates, to which are connected respectively the slit mechanism,
the prism box, and the plate holder, by three light steel castings,
forms the main body of the spectrograph. In the casting
to which the prism box is attached are mounted the collimator
and camera lenses, both of which are achromatic for the region
of 4500A. The spectrograph has an entirely new form of sup
port, designed by Campbell, to incorporate the suggestion made
136
THE BINARY STARS
by Wright, that such an instrument should be supported near
its two ends, like a bridge truss or beam, in order to give minimum
flexure. The support is a frame work of Tbars extending down
from the telescope, the form and arrangement of which are such
as to hold the instrument rigidly in the line of collimation of the
Plate IV. — The Mills spectrograph of the Lick Observatory.
large telescope. The lower support is a bar passing through a
rectangular opening in the casting carrying the prism box.
This bar is pivoted at the center of the casting and connected at
its two ends to the supporting frame. The upper support
consists of a cylindrical ring firmly attached to the frame work.
In this cylinder fits a spherical flange of the spectrograph casting,
the two forming a universal joint. Any strains originating
THE RADIAL VELOCITY OF A STAR 137
in the supporting frame cannot, with this form of mounting,
be communicated to the spectrograph. Careful tests of this
instrument and of the spectrograph of the D. 0. Mills Expedition
to Chile, which has the same form of mounting, show that the
effects of differential flexure have been eliminated. This method
of support permits, further, of a very convenient mode of moving
the spectrograph as a whole in order to bring the slit into the focal
plane of the large telescope, since it is only necessary to provide
sliding connections on the frame, for the lower support.
Nearly all modern stellar spectrographs are provided with
reflecting slit plates inclined at a small angle to the collimation
axis, which enable the observer to view the star image directly
on the slit. This is accomplished through the aid of a total
reflection prism, placed above the slit and outside of the cone
of rays from the telescope objective, which receives the light
from the slit and sends it to the guiding eyepiece. By placing
the slit parallel to the celestial equator, small errors of the
driving clock cause the star image to move along the slit, which
is desirable in order to obtain width of spectrum. Constant
and careful guiding is necessary to insure that the star's image
be kept exactly on the slit and that its motion along the slit be such
as to give a uniform exposure.
With a prism spectrograph and a straight slit the spectrum
lines are curved. The amount of the curvature depends upon
the optical constants of the instrument and the wave length
of the line. This source of trouble in measuring the spectro
grams may be eliminated for a short range of spectrum by
employing a slit of the proper curvature to make the spectrum
lines straight. Both threeprism instruments referred to above
are provided with curved slits.
As a source for the comparison spectrum, it is necessary to
select one giving a number of welldistributed lines in the part
of the spectrum to be studied. For example, for the new
Mills spectrograph in which the region 4400 to 4600A is utilized,
the spark spectrum of titanium is used. In the southern instru
ment, arranged for the region 4200 to 4500A, the comparison
source is the iron arc.
In order to eliminate the effects of any possible change in
the instrument during an exposure on the star, several impres
sions of the comparison spectrum are made at regular inter
vals. This is accomplished very conveniently and without
138 THE BINARY STARS
danger of changing the adjustment of the comparison apparatus
by a simple device due to Wright. Two small totalreflection
prisms are placed just above the slit, so that their adjoining
edges define the length of the slit. Two light sources are then
so arranged that the beam of each is brought to a focus on the
slit by a small condensing lens after total reflection in its respec
tive prism.
The optical parts of the spectrograph should, of course, be
mounted so that they cannot move, but care must be taken
that they are not cramped. This caution is especially pertinent
with regard to the large prisms. In the Mills spectrographs
the prisms rest upon hardrubber blocks and are firmly clamped
to one of the side plates of the prism box by light steel springs
which press against their upper surface. Small hardrubber
stops prevent lateral motion of the prisms.
In order to prevent the effects of changing temperature, the
principal parts of the spectrograph are surrounded by a light
wooden box, lined with felt. Over the felt surface are strung
a number of turns of resistance wire. The regulation of the
heating current is effected by means of a very sensitive mer
curyinglass thermostat by which the temperature inside of
the prism box is held constant during the night's work to within
a few hundredths of a degree centigrade.
The function of the telescope objective, for observations of
stellar spectra, is that of a condensing lens and the brightness
of the point image in the focal plane is directly proportional to
the area of the lens and its transmission factor. If we had
perfect seeing, we should receive in the slit of the spectro
graph, with the widths generally employed, about 90 per cent
of the light in the star image. Due to atmospheric disturbances
the image of a star under average conditions of seeing, is a
circular tremor disk whose diameter is four or five times the
width of the slit, so that the brightness of the spectrum is not
proportional to the area of the objective but more nearly to its
diameter. For example, the relative intensities of stellar spectra
obtained with the same spectrograph respectively upon the
36in. and 12in. refractors of the Lick Observatory would be
(allowing for the difference of transmission of the two), about
as two to one, since, for the photographic rays, the loss of light is
for the former about 50 per cent and for the latter about 25 per
cent. When a visual refractor is used for spectroscopic work,
THE RADIAL VELOCITY OF A STAR 139
it is necessary to render it achromatic for the photographic
rays. This is accomplished for the 36in. refractor by a correct
ing lens of 2j^in. aperture placed one meter inside the visual
focus of the telescope. This lens introduces an additional
loss of light of fully 10 per cent.
Since a silveronglass mirror has, under the best conditions,
a high reflecting power, and since it is also free from chromatic
aberration, it would seem that the reflector should be the more
efficient telescope to use in connection with a stellar spectro
graph. The reflector, however, possesses its own disadvan
tages, one of which is that it is very sensitive to changes of
temperature. Our experience with the 36in. refractor at
Mount Hamilton and the 373^in. reflector in Chile, when
used with high dispersion spectrographs, indicates that the
relative light efficiency of the two is about equal in the region of
H7. For apertures up to 36 in. one is inclined to favor the
refractor for high dispersion work, while for low dispersion,
where considerable extent of spectrum is desired, the reflector is,
of course, preferable.
The focal lengths of both refracting and reflecting telescopes
vary with change in temperature of the lens or mirror. It is,
therefore, necessary before beginning the night's work, and,
with the reflector, frequently during the night, to bring the
slit into the focal plane of the telescope, which, as noted above,
is effected by moving the spectrograph as a whole in the line
of collimation of the instrument.
It is well known that all high dispersion spectrographs are
very wasteful of light, though to what extent is perhaps not
always appreciated. When stellar spectrographs of three
prism dispersion are used in conjunction with large refractors
or reflectors, the combined instrument delivers to the photo
graphic plate probably less than 2 per cent of the light incident
upon the telescope objective. Half of the light is lost, as we
have noted, before it reaches the slit. The remaining losses
occur at the slit, in the prisms and in the collimator and camera
lenses of the spectrograph. In order to avoid unnecessary
losses of light, the obvious conditions must be satisfied, that
the angular apertures of the collimator lens and object glass are
the same, and that the prisms and camera lens are of sufficient
aperture to admit the full beam from the collimator. The
most serious losses occur at the narrow slit and in the prism
140 THE BINARY STARS
train. Indeed, one of the most important factors in the design
of stellar spectrographs, for maximum light efficiency, is the
proper balancing of these two conflicting elements, the trans
mission at the slit and the transmission of the prisms. Thus,
in the new Mills spectrograph, by using a collimator of slightly
greater focal length than the present one (28^ in.) with cor
responding increase in aperture of the lens and prisms, a wider
slit could be employed and still maintain the present purity of
spectrum. After allowing for the increased absorption of the
prism train, there would remain a small gain in light transmitted.
Although theoretically possible, this gain would probably be
more than offset by the inferior definition of the larger prisms and
the added difficulty of eliminating flexure. It is necessary here,
as at so many points in the spectrograph, to sacrifice a little
in order to gain more elsewhere. In fact, the most efficient
design of spectrograph may be described as the one in which
the wisest compromises have been made between the various
conflicting interests.
The decision as to the resolution and dispersion to be employed
is governed by several considerations : the type of stellar spectrum
to be studied, the size of the telescope at one's disposal, and
the brightness of the source whose spectrum can be photographed
with reasonable exposure times. With the spectrograph here
described two lines in the region of 4500A whose wave lengths
differ 0.2A are resolved, while the linear dispersion for 4500A
is 1 mm = 11 A. In order to obtain a spectrogram of suitable
density of a star whose photographic magnitude is 5.0, an
exposure time of an hour and a half is required. For stars of
photographic magnitudes 6.0 to 6.5 the width of the slit is
increased, thus sacrificing to some degree the purity of spectrum,
but not enough to interfere seriously with the accuracy of the
measures. In the case of early type stars whose spectra contain
single lines, the question of resolution is not important, and
where these lines are also broad, it is preferable to employ lower
dispersion. The adjustments of the various parts of the spectro
graph call for continual attention. It is necessary that the
instrument be placed with its axis of collimation accurately
in that of the large telescope and frequent tests should be made
to be sure that it remains so. The comparison source must be
adjusted so that its light follows very nearly the same path as the
starlight in the spectrograph. Care must be exercised at
THE RADIAL VELOCITY OF A STAR 141
every point in the process of obtaining and measuring the
spectrogram.
THE MEASUREMENT AND REDUCTION OF SPECTROGRAMS
For the measurement of spectrograms any one of the usual
forms of laboratory measuring microscopes will suffice. This
is merely a microscope on the stand of which is mounted a
carriage, movable by an accurate micrometer screw, in a direction
at right angles to that of the microscope axis.
In order to fix ideas, we shall assume that it is required to
measure and reduce a spectrogram of at Centauri, similar to the
one whose positive is reproduced in Plate III. The spectrogram
is first clamped on the carriage of the microscope, and the usual
adjustments of focus and alignment of the plate are made.
Great care should be taken that the illumination of the field
of the microscope is uniform. Beginning with the comparison
line 4250A, settings are made continuously along the plate on
good star lines and comparison lines as they chance to occur.
The plate is then reversed and the settings are repeated. It has
been shown by several investigators that the effects of errors
due to personal equation are practically eliminated by taking
the mean of the measures in the two positions. In tne reversal
of the plate the spectrum is also inverted, which may so change
the appearance of the lines as to interfere with the elimination
of personal equation. Especially is this true if the lines are
curved. The effects of accidental errors in setting are reduced
by employing a number of lines.
The table on page 143 contains the data of the measure and
reduction of this plate. Column I gives the wave lengths in
terms of the International Angstrom (I.A.) of the lines in the
iron comparison and the normal wave lengths of the star lines,
taken from the The Revision of Rowland's Preliminary Table
of Solar Wave Lengths. In columns IV and V are recorded
respectively the settings on the comparison and star lines (in
revolutions of the micrometer screw). The displacements
of the iron lines in the star are evidently given directly in amount
and sign by the differences, star minus comparison, and these
are entered at once in column VII (Displ.). We cannot enter
the displacements for the other star lines until the normal
positions of these lines have been obtained from those of the
142 THE BINARY STARS
iron comparison, by interpolation. This is effected in the
following manner:
A smooth curve drawn by plotting, for the comparison lines,
the reading on each line and its corresponding wave length,
respectively as ordinates and abscissae, will evidently repre
sent for this spectrogram the relation existing between wave
length and micrometer readings. From this curve — called a
dispersion curve — either the zero readings or the observed
wave length of the stellar lines could be obtained. This curve
was found by Cornu and later by Hartmann to be nearly of
the form of an equilateral hyperbola so that it is approximately
represented by the equation
X — X = z r (2)
A — Ao
where x is the micrometer reading on a line whose wave length
is X and Xo, x , and c are constants. Since it is not practicable
to plot the dispersion curve, the CornuHartmann formula
furnishes a very convenient means of obtaining it. The values
of the three constants are determined from three equations
formed by substituting the micrometer readings and wave
lengths of three lines, selected, one at each end of the region
of spectrum and the other near the middle. Micrometer read
ings of all other comparison and star lines are then computed
from the formula. The departure of this computed curve from
the true dispersion curve is furnished by a plot of the differ
ences between the observed and calculated readings of the
comparison lines. The computed normal positions of the star
lines are then corrected for the difference between the com
puted and observed dispersion curve. The decimal portions of
the results would be entered in column VI (Sup'd).*
As before, the difference, star line minus zero line, gives the
displacement in revolutions of the screw. In order to express
this as AX, that is in units of wave length, it is necessary to
know r, the number of Angstrom units in one revolution of the
screw. The value of r for any point in the spectrum is evi
dently the slope of the dispersion curve at that point, and is
equal to (X — X )/(x — x ). Finally, in accordance with the
* The figures actually entered in this column in the example were obtained
by a different method of reduction which is explained in the paragraphs
following.
THE RADIAL VELOCITY OF A STAR
143
* az Centauri ft.
* Plate No. 3791 III
ol4 h 32.8 m
Date 1911 Feb. 27
X
Table
Oo
Ta
Comp.
*
Sup'd.
Displ.
TVs
v,
4250.132
54.886
54.886
54.758
0.128
319
40.8
4250.799
55.031
3
55.034
54.909
0.125
320
40.0
4282.413
61.819
13
61.832
61.710
0.122
335
40.9
4283.016
61.944
61.831
958
0.127
335
42.5
4294.147
64.250
16
64.266
64.140
0.126
338
42.6
4299.252
65.295
20
65.315
65.190
0.125
340
42.5
4312.877
68.039
67.944
061
0.117
349
40.8
4313.633
68.190
68.090
212
0.122
349
42.6
4318.660
69.185
69.105
220
0.115
352
40.5
4325.000
70.431
70.355
469
0.114
356
40.6
4325.777
70.584
40
70.624
70.502
0.122
356
43.4
4327.919
71.001
70.928
041
0.113
357
40.3
4337.057
72.767
43
72.810
72.692
0.118
360
42.5
4340.477
73.421
73.350
467
0.117
362
42.4
4359.625
77.027
76.970
082
0.112
372
41.7
4369.781
78.896
78.844
957
0.113
376
42.5
4375.946
80.018
79.972
083
0.111
378
42.0
4379.240
80.612
80.571
680
0.109
380
41.4
4383.559
81.388
70
81.458
81.352
0.106
382
40.5
4399.780
84.257
84.228
337
0.109
390
42.5
4404.763
85.126
86
85.212
85.105
0.107
392
41.9
4406.654
85.453
85.432
539
0.107
394
42.2
4415.137
86.913
93
87.006
86.898
0.108
397
42.9
4425.446
88.664
88.662
759
0.097
402
39.0
4428.551
89.198
89.194
296
0.102
404
41.2
4430.624
89.536
89.535
636
0.101
404
40.8
4434.969
90.270
90.270
372
0.102
406
41.4
4435.690
90.380
90.378
482
0.104
406
42.2
4442.351
91.482
108
91.590
91.491
0.099
411
40.7
4443.814
91.724
91.732
831
0.099
412
40.8
4447.730
92.365
92.375
473
0.098
413
40.5
4459.140
94.216
114
94.330
94.238
0.092
417
38.4
4476.023
96.906
127
97.033
96.940
0.093
426
39.6
4482.217
97.872
131
98.003
97.905
0.098
428
41.9
4494.575
99.782
138
99.920
99.820
0.100
434
43.4
 35)
1449.9
41
.43
Scs
lie = +0
va = +21
vd = 
.13
.82
.07
Obse
jrved V
19
55 km
144 THE BINARY STARS
relation deduced on page 126, v the observed radial velocity is
obtained by multiplying AX for each line by its corresponding
factor V. = 299,796/X.
Each spectrogram may be reduced in the manner outlined
above, and some observers prefer to follow this method rigor
ously for each stellar spectrogram. When this is done the
process is simplified by carrying through the computation in
wave lengths, so that the displacement is expressed at once
in Angstroms.
Since for the same spectrograph the form of the dispersion
„urve differs but slightly for different temperatures, a simple
and practical method of reduction is offered by the following
procedure: A standard dispersion curve is computed once for
all, according to the method described above, from measures
of a solar spectrogram. With the aid of this all other spectro
grams taken with the spectrograph may be quickly and easily
reduced. It is convenient to put this standard curve in the
form of a dispersion table in which are entered the normal wave
lengths of the comparison and stellar lines used for stars of differ
ent spectral classes, and the micrometer readings corresponding
to these wave lengths. In this standard table are given also the
values of rV, for each line. Columns I, II, and VIII, in the
example, are taken from such a table.
It is now only necessary to reduce the readings of the standard
table to the dispersion of the plate, by plotting the differences
between the observed and table readings of the comparison
lines (recorded in column III in the example). From this
curve the difference to be applied to the table reading for each
star line is read off. In the sixth column are given the new
table readings (for zero velocity) after this difference has been
applied. When there are comparison lines corresponding to
star lines some observers follow rigorously the process outlined,
while others (as in the present example) take the difference
between the readings of the two as the displacements. The
last three columns contain, respectively, the displacements
(* minus Comp. or Sup'd), the factor rV S) and the products
of these two values, which are the relative radial velocities of
star and observer as supplied by the lines measured. The
mean of the measures for 40 lines gives as the observed radial
velocity —41.43 km/sec. It will be noticed that the dispersion
of the star plate is about 0.3 per cent greater than that of the
THE RADIAL VELOCITY OF A STAR 145
standard table, and consequently the factor r (computed for the
table) is too large, and the numerical value of this velocity must
be reduced by this amount. This is allowed for, in the example,
as scale correction. In practice, it is convenient to have several
standard tables corresponding to the dispersion of the spectro
graph at different temperatures. The one whose dispersion is
nearest that of the star plate is selected for use. Experience has
shown that the results obtained by the very simple method
just described are of the same accuracy as those derived by
the longer process of computing a dispersion curve for each
plate.
If the spectrograph is not provided with a curved slit, it is
necessary to introduce a correction for the curvature of the
lines. This correction may be computed from Ditscheiner's
formula* or determined empirically from lines on a spectrogram
of the Sun, on the assumption that the curve of each line is a
parabola. The better method is to eliminate the source of
this correction by the use of a curved slit.
The observed radial velocity of a star is made up of the star's
velocity, V, with reference to the solar system, and the velocity
of the observer in the solar system. The latter consists of three
components, which arise from (1) the revolution of the Earth
around the Sun; (2) the rotation of the Earth on its axis; (3)
the revolution of the Earth around the center of mass of the
EarthMoon system. This last component never exceeds
±0.014 km/sec and may be neglected. The correction for the
annual and diurnal motions of the Earth are readily computed
from the formulas given by Campbell in FrostScheiner's Astro
nomical Spectroscopy (pages 338345). The values for these
in the example are given respectively under v a and v d . Hence,
the observed radial velocity of a 2 Centauri with reference to the
Sun on 1911, February 27.883 (Greenwich Mean Time) was
— 19.55 km/sec.
Methods of reduction which depend upon dispersion for
mulas require an accurate knowledge of the wave lengths of
the lines used in both the comparison and stellar spectra. Accu
rate values of the absolute wave lengths are not required but
their relative values must be well determined. For example,
* Uber die Krummung der Spectrallinien, Sitz. Ber. d. Math. Klasse d. k.
Akad. Wien, Bd. LI, Abth. II, 1865; also FrostScheiner, Astronomical
Spectroscopy, p. 15, 1894.
146 THE BINARY STARS
a relative error of +0.01 A in the wave length of any line would
produce an error in the velocity for that line of nearly a kilometer.
Interferometer measures of the wave lengths in the spectra
of a number of elements are now available, but for the wave
lengths of solar lines it is still necessary to use the determinations
by Rowland. Fortunately, we now possess accurate wave
lengths of the lines in the spectra of most of the elements, and the
Revision of Rowland's Preliminary Table of Solar Wave Lengths
has furnished a homogeneous set of data for the lines of the
Sun's spectrum. A serious difficulty, however, arises for stellar
lines, from the fact that stellar spectrographs have not sufficient
resolution to separate lines which were measured as individual
lines in the solar spectrum with the more powerful instruments
employed by Rowland and his successors. It is the practice of
many observers, where two lines merge to form one line in the
star spectrum, to take the mean of the wave lengths of the
component lines, weighted according to the intensities given by
Rowland for those lines in the Sun. Wave lengths based on
estimates of intensity should naturally be regarded with suspicion,
and in fact we do not know, until the entire plate has been
reduced, whether we have chosen an erroneous wave length or
not. It is well known that various stellar lines and blends behave
differently for stars of different types. The lines in solar type
stars are assumed to have the same wave lengths as similar
lines in the Sun. In the case of stars of other spectral classes,
the solar lines which occur can be used in determining the
wave lengths of the nonsolar lines and blends. In this manner
special tables are constructed for stars of different types.
When spectrographs of lower dispersion and resolution than
those of three prisms are employed for the measurement of solar
and latertype spectra, the effect of uncertainties in wave length
of the stellar lines, due to blends, becomes very serious. The
two methods of measurement and reduction which follow
eliminate the sources of error incident to the use of blends,* and
erroneous wave lengths as far as it is possible to do so. The
first is that due to Prof. R. H. Curtiss and is called by him
the velocity standard method. In principle it amounts to a
determination of the wave lengths of the lines in the spectrum
of a source whose radial velocity is known made with the particu
lar spectrograph which is to be used for measures of stellar spectra
of this same class. Thus for the measures of spectra of the
THE RADIAL VELOCITY OF A STAR 147
solar type, a table similar to the one we have described is formed.
The micrometer readings in this table, however, are not com
puted from assumed wave lengths, but are the mean of the actual
settings, on comparison and solar lines, obtained on several
spectrograms of the Sun. These standard plates are produced
as nearly as possible under the same conditions as the stellar
plates to be measured. The procedure in the reduction of the
measures by means of this table is then the same as that described.
It is necessary, of course, to correct the measured stellar velocity
for the radial velocity of the source when the standard spectro
grams were taken. Standard tables for the reduction of measures
of stars of other spectral classes may be formed in a similar
manner, using as the standard sources stars whose radial veloci
ties are well determined.
The second method is due to Prof. Hartmann and is in prin
ciple the same as the preceding one, except that the star plate
is referred directly to the standard plate on a special measuring
microscope, known as the spectrocomparator. The instru
ment is provided with two plate carriages, one of which is
movable. On one of the carriages the star plate is placed
and on the other, which is provided with a fine micrometer
screw, is a standard plate of the Sun (taken with the stellar
spectrograph). The microscope has two objectives so arranged
that the images of portions of the two plates are brought, by
means of total reflection prisms and a reflecting surface, to
focus in the same plane and in the field of one eyepiece. By
means of a silvered strip on the surface of one prism, the central
portion of the Sun's spectrum is cut out and the star spectrum
thrown into its place. In a similar manner, central strips of the
comparison spectra of the Sun plate are replaced by those of
the comparison spectra of the star plate. An ingenious arrange
ment of the microscopes permits of equalizing the scale of the
two plates, by changing the relative magnifying powers of the
two objectives. The method of measurement is, then, after
proper alignment of the plates, to bring corresponding sections
of the two plates into the field of the microscope, and by means
of the micrometer screw set the corresponding lines of the
comparison spectra in the same straight line. A setting is then
made with the corresponding lines of the solar and star spectra
in the same straight line. The difference between the micrometer
readings in the two positions is the displacement of the star lines
148 THE BINARY STARS
relative to the solar lines. In practice it is found sufficient to
divide the length of the spectrum into about 15 sections, for each
of which these comparative settings are made. The mean of
the displacements, obtained with the plates in the direct and
reverse positions, when multiplied by the rV 8 for each section,
gives for each the value V* — V , where V* is the radial velocity
of the star and T that of the Sun. Theoretically, the values
of V* — V Q should receive weights proportional to 1/rV, in
taking the mean. Although this correction is negligible, except
where an extent of spectrum of 400 or 500A is used, its introduc
tion leads to a very simple method of computation. Take the
sum of the displacements in the direct and reverse measures
and multiply by a factor / = — • The product is equal to
the weighted mean of the values V*  V for each section.
This, corrected for the velocity of the original Sun plate (Fo),
gives the radial velocity of the star relative to the observer.
The reduction to the Sun is made in the usual way. The
factor / is a constant so long as the same regions are used, and its
values may be computed for all combinations of the regions that
are used. The great advantage of the method, aside from those
which it possesses in common with the velocity standard method,
is that we are able to measure and reduce in an hour a plate of a
star rich in lines, and practically utilize all the material on the
plate. With the older methods, to make such a comprehensive
measure and reduction, i.e., to utilize all of the lines on the plate,
would require one or two days.
For the measures of spectra of a type other than the solar
it is necessary to select for the standard plate a spectrogram
of a star of that particular spectral class. In order to obtain
the velocity for this standard spectrogram, it should be meas
ured and reduced, either by the method first described or perhaps
preferably by the velocity standard method. The adopted
value should be the mean of the measures made by several
different observers.
The spectrocomparator offers a very efficient method for
determining the differences in velocities of the same star, by
measuring a series of plates of the star with reference to one
of these selected as a standard.
THE RADIAL VELOCITY OF A STAR 149
Five of the six elements of a spectroscopic binary orbit depend
only upon the accurate determination of the relative radial veloci
ties given by the series of spectrograms. One of the most
important applications of the Hartmann comparator is, therefore,
to the measurement of plates of a spectroscopic binary.
For the measure and reduction of spectrograms of stars of
the earlier spectral classes, the use of the CornuHartmann dis
persion formula will suffice, inasmuch as the spectra of such
stars consist of lines due to the simple gases, the wave lengths
of which have been accurately determined in the laboratory.
The measure and reduction of spectrograms of stars of the
solar and later classes of spectra are accomplished with great
saving of time and labor, and by a method free from some of
the uncertainties of wave lengths, by the use of the spectro
comparator. If the observer is not provided with such an
instrument the velocity standard method is preferable to the
use of the dispersion formulas, at least until a system of stellar
wave lengths of the requisite accuracy is available.
To the reader who has followed the long and intricate process
of determining the radial velocity of a star, the question will
naturally occur, how do we know that the final result repre
sents the star's velocity? Obviously, the final test of the
method is its ability to reproduce known velocities. Fortu
nately, we have at hand a means of making such a test. Since
the orbital elements of the inner planets of the solar system
are well determined, we can readily compute the radial velocity
of one of these with reference to the Earth at any given time.
It is only necessary, then, to observe the relative radial velocity
of the planet and the Earth and compare this with the computed
value at the time of observation. At the Lick Observatory
spectrograms of Venus and of Mars are secured at frequent
intervals with the stellar spectrograph and measured by the
observers in the regular course of measuring stellar plates.
With the threeprism spectrograph, described above, the observed
and computed velocities of these two planets generally agree
to within ±0.5 km, or the unavoidable error of measure. When
the spectrograms are measured by several observers, the effects
of personal equation are to some extent eliminated in the mean,
and an agreement within a few tenths of a kilometer is to be
expected. A continual check is thus afforded on the adjustments
of the spectrograph and the measurement of the spectrograms.
150 THE BINARY STARS
References
General
Campbell, W. W.: Stellar Motions, Yale University Press, 1913.
Eberhard, G.: Sternspektrographie und Bestimmung von Radialge
schwindigkeiten, Handbuch der Astrophysik, I, chap. 4, 1933.
Instruments and Design
Campbell, W. W.: The Mills Spectrograph, Ap. Jour. 8, 123, 1898.
Frost, E. B., The Bruce Spectrograph, Ap. Jour. 16, 1, 1902.
Hartmann, J.: Remarks on the Construction and Adjustment of Spectro
graphs. Ap. Jour. 11, 400, 1900; 12, 31, 1900.
Keeler, J. E. : Elementary Principles Governing the Efficiency of Spectro
graphs for Astronomical Purposes, Sidereal Messenger 10, 433, 1891.
Newall, H. F. : On the General Design of Spectrographs to be Attached to
Equatorials of Large Aperture, Considered Chiefly from the Point of
View of Tremordiscs, Mon. Not. R.A.S. 65, 608, 1905.
Plaskett, J. S. : Spectrograph of the Dominion Astrophysical Observatory,
Publ. Dominion Astroph. Obs. 1, 81, 1920.
Vogel, H. C: Description of the Spectrographs for the Great Refractor at
Potsdam, Ap. Jour. 11, 393, 1900.
Wright, W. H. : Description of the Instruments and Methods of the D. 0.
Mills Expedition, Publ. Lick. Obs. 9, 25, 1905.
Methods op Measurement and Reduction
Campbell, W. W. : The Reduction of Spectroscopic Observations of Motions
in the Line of Sight, Astronomy and Astrophysics, 11, 319, 1892. Also,
FrostScheiner, Astronomical Spectroscopy, p. 338.
Curtiss, R. H. : A Proposed Method for the Measurement and Reduction of
Spectrograms for the Determination of the Radial Velocities of Celestial
Objects, Lick Obs. Bull. 3, 19, 1904; Ap. Jour. 20, 149, 1904.
Hartmann, J. F.: Uber die Ausmessung und Reduction der Photograph
ischen Aufnahmen von Sternspectren, A.N. 156, 81, 1901.
: A Simple Interpolation Formula for the Prismatic Spectrum, Ap.
Jour. 8, 218, 1898.
: The Spectrocomparator. Ap. Jour. 24, 285, 1906; Publ. Astroph.
Obs., Potsdam, 18, 5, 1908.
[For a modern discussion of spectrum analysis see : J. A. Hynek, ed., Astro
physics, pp. 12258, 1951; R. H. Garstang, Peculiar Stars, R. A. S. Occasional
Notes 3, 21, Nov. 1959; and Helmut A. Abt, A Discussion of Spectral Classi
fication, Ap. Jour. Supplement Series 8, 75, Apr. 1963. — J.T.K.]
CHAPTER VI
THE ORBIT OF A SPECTROSCOPIC BINARY STAR
The problem of determining the orbit of a binary system
from measures of radial velocity, made in the manner described
in the previous chapter, differs in several important particulars
from that of computing an orbit from micrometric measures of
position angle and distance. It has been shown that microm
eter measures provide the data from which the projection of
the orbit of the companion star with respect to its primary can
be drawn, the true relative orbit following, correct in propor
tions but of unknown linear dimensions. The radial velocities,
on the other hand, when plotted against the times, produce a
periodic curve, having the general appearance of a distorted
sine curve; from this curve we are to find the elements of the
true orbit of the star with respect to the center of gravity of
the system of which it forms one component.*
Figure 9 illustrates the conditions of the problem. Let the
.XTplane be tangent to the celestial sphere at the center of
motion, and let the Zaxis, perpendicular to the XTplane, be
parallel to the line of .sight along which the radial velocities
are measured. The velocities are considered positive (+) when
the star is receding from, and negative ( — ) when it is approaching
the observer. The orientation of the X and Faxes remains
unknown. Let PSA be the true orbit of the star with respect
to the center of motion and let the orbit plane intersect the
ZFplane in the line NN'.
Then, when the star is at any point S in its orbit, its distance
z from the .XTplane will be
z = r sin i sin (v + «)
* It is here assumed that the spectrum of only one component is visible;
when both components give spectra, we may determine the relative orbit
of one with respect to the other, using the same formulas but changing the
value of the constant of attraction. The relative and absolute orbits are,
of course, similar in every respect.
151
152
THE BINARY STARS
the symbols in the righthand member of the equation having
the same significance as in the case of a visual binary star.
The spectrograph, however, does not give us the distances of
the star from the .XYplane, but the velocities of its approach
Fig. 9. — The spectroscopic binary star problem.
to, or recession from this plane, generally expressed in kilometers
per second. The radial velocity at point S is equal to dz/dt,
and is therefore expressed by
dz . . . . . .dr . . . , . .dv
37 = sin 1 sin (y + «) 77 + r sin t cos (v + w)j
From the known laws of motion in an ellipse we have
dr _ fiae sin v
dt
dt
and therefore
dz
dv _ m<*(1 + e cos v)
vr=
vr=
na sin 1
dt y/\ 
[e cos co f cos (v + «)]
(1)
which is the fundamental equation connecting the radial velocities
with the elements of the orbit. *
* In place of (v + w) the symbol u ( = the argument of the latitude) is
dr du
often used, the expressions for n and r^7 written
dt
dt
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 153
The observed velocities evidently contain the velocity, V,
of the center of mass of the system, which is a constant quan
tity for any given simple binary system,* as well as the variable
velocities due to the star's orbital motion and the quantity V
must therefore be subtracted from the observed values to make
them purely periodic. In other words, the velocity curve is
purely periodic only with respect to a line representing the
velocity of the system as a whole. This line is called the Faxis.
Equation (1) applies only to the velocities counted from the
,7axis. If dt/dt represents the velocity as actually observed
(i.e., the velocity referred to the zeroaxis) we shall have the
relation f
f= F + ^ = F + J^±[ecos (a + cos (v + «)]. (la)
dt dt "vl — e 2
Methods of determining the position of the Faxis will be
given later; for the present we shall assume it to be known.
Five constants enter the right hand member of Eq. (1), viz.,
a sin i, e, n, co, and (through v) T. These express the five orbit
elements which it is possible to determine by measures of radial
velocity.
Since the inclination of the orbit plane is not determinable,
the value of a, the semimajor axis, must also remain unknown.
It is therefore customary to regard the function a sin * as an
I^» to <»— >• and ^ / 7 5 = 7j 11+eeos( "" ,)
and hence the fundamental equation in the form
dz I ■ • / i \
j = — =. sin % (cos u + e cos w).
dt ^Tp
In these equations p [ = o(l  e 2 )] is the semiparameter of the true
ellipse and / denotes the constant of attraction, which, when the spectrum
of only one component is visible, and the motion is determined with refer
ence to the center of mass of the system, takes the form kmfi/{m + mi),
k being the Gaussian constant; when both spectra are visible and the motion
of one star with respect to the other is determined, / = ky/m + mi It is
clear that the form of the fundamental equation will be the same whatever
value we may assign to / and the constant of attraction may therefore be
disregarded until the question of the relative masses in the system comes
up for discussion.
* In a triple or multiple system, this quantity will itself be variable,
f The symbol y is often used for the velocity of the system instead of V.
154 THE BINARY STARS
element. Further, it is clear that the position of the line of
nodes cannot be determined, though we can find the times when
the star passes through each of the nodal points. The various
elements have the same definitions as in the case of visual
binary star orbits (see page 78), except that the angle « in spec
troscopic binary orbits is always measured from the ascending
node, the node at which the star is moving away from the observer.
It will be seen later that the radial velocity has its maximum
positive value at this node and its minimum positive value (or
maximum negative value) at the descending node. It should
also be noted that the unit of time for fi (and therefore for P)
is the day, not the year as in visual binary orbits.
Theoretically, values of the radial velocity at five different
times suffice for the complete solution of Eq. (1); practically,
no computer undertakes an orbit until a considerable number of
measures is available which give the velocities at short intervals
throughout the entire revolution period. To secure a satis
factory distribution of the observations a preliminary value
of the period is necessary and such a value can ordinarily be
obtained without difficulty by plotting the early observations
on coordinate paper, taking the times, expressed in Julian
days and decimals of a day, as abscissas and the velocities,
expressed in kilometers per second, as ordinates. A convenient
epoch as origin for the period is selected near the beginning of
the series, preferably one corresponding to a point of maximum
or minimum velocity. If later measures indicate that the
period is in error, a new period which is a submultiple of the
original one will often prove satisfactory. In difficult cases,
the following artifice may be found helpful.* Copy on trans
parent paper from onethird to onehalf of the series of observed
points, choosing the time interval best covered by observa
tion ; slide the copy along the original plot, keeping the time axis
in coincidence, until some point on the copy falls approximately
upon a different point in the original at which the velocity is
changing in the same direction. The time interval between the
* This was suggested to me by Dr. R. K. Young who said that it had been
used with good results by several computers of binary star orbits at the
Dominion Observatory. No prior mention of the device has been found in
print and its author is unknown to me. Its usefulness arises from the fact
that, in effect, it doubles the number of observations for a given time
interval.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 155
two points is evidently equal to the period or a multiple of the
period.
Schlesinger* has published a criterion that may be applied
to advantage in cases where an observer has accumulated
many plates of a star which apparently shows variable radial
velocity without being able to determine any period. It con
sists in constructing a frequency curve for the velocities by
"dividing the total range exhibited by the measured velocities
into successive groups of equal extent, say 3 km each, and
then counting the number of velocities that fall within these
groups. Regarding these numbers as ordinates, we plot them
and join the ends by a smooth curve." This curve is com
pared with the wellknown error curve; if the two are the same,
within reasonable limits, we may conclude that the differences
in the measured velocities are due to errors of observation,
and afford no support for the assumption that the star is a
spectroscopic binary. If the two curves differ, the star is a
binary and the form of the frequency curve will give an idea
as to the general character of the orbit and frequently furnish a
clew to the period. For Schlesinger shows that circular orbits,
elliptic orbits with periastron at descending node, elliptic orbits
with periastron at ascending node, and elliptic orbits with
periastron removed 90° from the nodes, all have characteristic
frequency curves which differ in form from the error curve.
When the nature of the frequency curve has shown to which of
these classes the orbit in question belongs, it becomes very
much easier to decide upon the epochs for the various observed
velocities, and thus upon an approximate value for the period.
When the period is approximately known all of the observa
tions may be reduced to a single revolution by subtracting
* Ap. Jour. 41, 162, 1915. In his paper on the "Orbit of the Spectro
scopic Binary x Aurigae" (Jour. R.A.S.C. 10, 358, 1916), Young shows that
the errors of measurement may affect the expected distribution in such a
manner as to mask to a considerable degree the presence of the orbital
variation. If possible, all of the spectrograms used in the determination
of a particular orbit should be secured with the same instrument and meas
ured by the same person, to avoid the effect of systematic errors of observa
tion and of the personal equation in measurement. The small systematic
differences between the radial velocities of stars made at different observa
tories have been discussed by several investigators. See, for example, Dr.
J. H. Moore's Introduction to his General Catalogue of the Radial Velocities
of Stars, Nebulae, and Clusters (Publ. Lick Obs. 18, 1932).
156 THE BINARY STARS
multiples of the period from the later dates. A preliminary
curve is drawn to represent the plotted positions as closely as
possible. The deviations from the curve at points near the
mean of the maximum and minimum velocities, where a change
in the periodic time will have the greatest effect, will indicate
advisable changes in the assumed period and these are readily
found by dividing the deviations of such critical observations,
expressed in time, by the number of revolutions elapsed. A
second curve is then drawn whose periodic time will generally
be very close to the true value. In practice it will frequently
happen that two or three measures of the radial velocity of a
star are available which were made (perhaps at another obser
vatory) several years before the series of spectrograms for the
orbit computation is begun. When a fairly good value of
the period has been found from the later series, these early plates
will determine its true value with high precision. Generally
they are not used in finding the other orbit elements.
When the period has been determined as accurately as possible
and a series of spectrograms has been accumulated giving
the velocities at points well distributed throughout the entire
period, the most probable curve is drawn, by estimation, through
the points as plotted, and, if the ingenious methods of super
position devised by Schwarzschild and Zurhellen are to be
used, the curve should be prolonged through a revolution and
a half. The plotted points used for this curve should repre
sent normal positions, formed by combining several velocities
observed at very nearly the same orbit phase, whenever the
number of observations is sufficient to permit such combina
tions. In making the combinations, the question of weights
arises, and here the practice of computers varies considerably,
for several factors enter. The character of the lines on the
spectrograms, broad or narrow, sharp or illdefined, strong or
weak, is one factor; the number of lines is another; if the plates
have been taken with different telescopes and spectrographs,
a third factor is introduced. These must all be considered
in assigning the weights to each plate. The only direction
that can be given is the general one to use a rather simple system
of weighting. It will rarely be of advantage to assign fractional
weights, or to use a range of weights greater than, say, four units.
The weights should, of course, be assigned to each plate, at the
time of measurement.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 157
The errors in drawing the most probable curve have con
siderable effect upon the accuracy of the determination of the
elements. At best the curve is not likely to be a perfect repre
sentation of the elliptic motion which caused it since it is natural
to bend the curve slightly in or out at different points to satisfy
the more or less exact observations. This difficulty is inherent
and for it there is apparently no remedy other than that of
testing the first orbit by a trial ephemeris and making the small
changes in the elements which are indicated by the residuals.*
OKm.
Zero
axis
10 2030405060 70 80 90 100 U0120c*gu
6476.0 U U
Fio. 10. — Velocity curve of k Velorum.
If Fig. 10 represents a velocity curve, it is evident from
Eq. (1) that the points A and B correspond respectively to
the ascending and descending nodes of the star's orbit, for
at the times of nodal passage we have (v + «) = 0°, and
(v + a>) = 180°, respectively, and therefore cos (v + «) = ±1.
The radial velocity thus reaches its maximum and minimum
values at the nodal points.
Taking A and B as the magnitudes of the curve ordinates
at the points of maximum and minimum reckoned from the
7axis, regarding B a s a po sitive quantity and writing for
brevity K = na sin ijy/\ — e 2 , we have
A = K(l + e cos «)
B = K(l — e cos w)
* King's method affords a graphical test of the first orbit found, see page
170.
158 THE BINARY STARS
and therefore
A +B
2
A B
2
A ~ B
= K
— Ke cos co } (2)
= e cos w
A + B
Hence we may write Eq. (1) in the form
■£ = K[e cos co + cos (v + co)] = — 1 ±— cos (v + co) (3)
and Eq. (la) in the form
^ = K H ^ 1 2 — C0S ( w + w ) = ^i +
— 2 — cos (t; + co). (3a)
if is therefore the half amplitude of the velocity curve.
Up to the point now reached practically all methods of spec
troscopic orbit determination are identical. But when the
fundamental relations are given as above, and the curve has
been drawn, various methods are available for computing the
orbit elements, other than the period, which is assumed to be
known.
Of these, the method devised by LehmannFilhSs will first
be presented, essentially in full; other methods will then be
treated in less detail. The student who desires to study the
various methods more fully is referred to the important papers
given in the references at the end of the chapter.
METHOD OF LEHMANNFILHES
Given the observations, and the velocity curve drawn with
the value of P assumed as known, the first step is to fix the
Faxis, the line defining the velocity of the center of gravity
of the system. This is found by the condition that the integral
of dz/dt, that is, the area of the velocity curve, must be equal
for the portions of the curve above and below the 7axis. By
far the easiest method of performing this integration is to use
a planimeter. A line, approximately correct, is first drawn by
estimation; the areas contained between it and the curve above
and below are measured, and the difference between the two
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 159
is taken. The position of the axis is then shifted to eliminate
this difference, and the measures are repeated. It will rarely
be necessary to make more than one correction to secure an
accurate value of the position of the Faxis, which, by this
method, depends upon the entire curve.
If a planimeter is not available, the areas above and below
the axis may be equalized by using coordinate paper for the
plot of the curve and counting the small squares in each area.
Approximate mechanical integration, as advised by Lehmann
Filh6s, may also be resorted to, by those who enjoy this form
of recreation.
Having found the Faxis, the ordinates to it are next drawn
from the points of maximum and minimum velocity, A and B.
It is at this point, as Curtis says, that the method is weakest,
for slight errors in fixing the position of A and B may easily
arise. It is well to apply the check afforded by the requirement
that area AaC (Fig. 10) must equal CbB and DaA equal BID.
Since C and D lie on the 7axis the velocities at these points
are zero, hence from Eqs. (3) and (2) we have for dz/dt at these
points
cos (v + w) =  A , B = ~e cos co (4)
If Vi is the true anomaly corresponding to the point C, which
is traversed by the star on the way from the ascending to the
descending node, and « 2 , the true anomaly for the point D,
sin (vi + «) will be positive, sin (v 2 + «) negative, and we
shall have
A  B , . x A  B )
cos (t>i + w ) =  A + B > cos (v* + »;   A + 5 
sm (»! + «) = A+B > sm («* + «) = ~ A + b
Let Zi and Z 2 denote the areas* AaC and bBD (Fig. 10), respec
tively, and let n and r 2 be the radii vectores for the points C
and D.
Then
Z x = r x sin i sin (v x + o>)
Z 2 = r 2 sin i sin (i> 2 f <a) = r 2 sin i sin (vi + co)
* These areas represent the distances of the star from the XFplane at the
points in its orbit corresponding to (»i + «) and (» 2 + »)•
(5)
160 THE BINARY STARS
and therefore
— — \ =  1 = 1 + g COS V 2
Z 2 r 2 1 + e cos Vi ^ '
since r = [a(l  e 2 )]/(l + e cos 0). Write (v + w  &>) for v,
in Eq. (6), expand, and reduce, with the aid of the relations in
Eqs. (5) and (4), and we have
_Zi sin (vi + w) — e sin a>
Z 2 sin (t>i + w) + e sin w
whence
Equation (7) and the last of Eq. (2) determine e and o>. The
values of A and B are taken from the curve, and the areas Z\
and Z 2 are quickly integrated from the curve portions AaC
and bBD by means of a planimeter, the latter area being regarded
as negative in sign. Since the areas enter as a ratio, the unit
of area used is entirely immaterial.
At the time of periastron passage v — 0°; hence from Eq. (3)
we have
£ = KQ. + e) cos a, (8)
which gives the ordinate corresponding to the point of peri
astron passage. Two points of the curve will have the same
ordinate, but since (v + o>) equals 0°, 180°, and 360° for the
points A, B, and Ai, respectively, there will be no ambiguity
as to the position of the periastron point. The abscissa of
this point, properly combined with the epoch chosen for the
beginning of the curve, defines T, the time of periastron pas
sage. Instead of using Eq. (8) we may find T by determining E
for the point C for which the value of v is known, and then
employ the formulas
tan }^E = J e tan }4v)
V +e C (8a)
T = i E ~ e sin E
or, if the eccentricity is less than 0.77, the value of M correspond
ing to v may be taken directly from the Allegheny Tables, and T
found from the relation
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 161
M = n(t  T). (86)
Such procedure is especially advisable when the periastron
points fall near point A or B on the curve.
By definition (page 157) we have
fj.a sin i
K. —
and hence
A_+ B V 1  e 2
a sin i 2 a sin i
from which we may find the value of the product a sin i. Since
the unit of time for A and B is the second, while for m it is the
day, the factor 86,400 must be introduced. Our equation
then becomes
a sin i = 86,400  \/T^7 2 = (4.13833)KP y/\  « 2 (9)
the number in parentheses being the logarithm of the quotient
86400 s 2tt.
Summarizing, the practical procedure is:
1. Find the period as accurately as possible by successive
trials and plot the most probable velocity curve on the basis of
normal places.
2. Find the position of the 7axis by integration of areas, using
the planimeter, if available.
3. Measure the ordinates for points A and B and find the areas
of AaC and bBD expressed in any convenient units.
4. From Eqs. (2) and (7) determine K, e, and <i.
5. From Eq. (8), or by calculation from the value of v, for
the point C, determine T.
6. From Eq. (9) determine a sin i.
To test the elements by comparison with the observations,
we compute the radial velocity for each date by the formulas:
M = fx(t  T) = E  e sin E
tan Y 2 v = Jr— tan % E
$ = V + Ke cos to + K cos (v + «)
at
(10)
162
THE BINARY STARS
The value of v for each value of M may be taken directly from
the Allegheny Tables, if e is less than 0.77.
To illustrate LehmannFilheV method I have chosen the orbit
computed for k Velorum, by H. D. Curtis, the velocity curve
for which is given in Fig. 10.
The observations used were as follows:
Julian Day, G. M. T.
Velocity
Julian Day, G. M. T.
Velocity
2,416,546.739
+68. 5^
_ _
2,417,686.591
+33. 8*™
60.703
+12.9
91.572
+38.2
97.651
+65.7
92.545
+43.2
6,912.601
+53.3
96.480
+46.7
7,587.844
+58.6
7,701.494
+52.7
88.788
+57.9
41.466
+22.1
90.829
+58.5
46.463
+ 0.3
91.824
+64.8
49.470
 7.6
97.788
+65.8
50.479
 8.8
7,609.790
+62.0
51.463
13.3
54.534
21.0
53.457
19.2
55.556
19.2
58.451
29.0
58.570
15.2
59.460
24.6
59.545
14.5
The small circles representing the first four observations,
which are important in determining the period, owing to their
distance in time from the later ones, are barred in the diagram.
The period, P, was assumed to be 116.65 days, and the begin
ning of the curve is at Julian day 2,416,476.0. This is not
exactly at a minimum, as may be seen from the diagram.
From measures of the curve we find
A = 46.3
B = 46.9
A + B = 93.2
A  B = 0.6
K = (A + B)/2 = 46.6
Zi = AaC = +0.168
Z 2 = bBD = 0.259
Z 2 + Zx = 0.091
Z 2  Zt 0.427
the figures for area being expressed in decimals of the unit of
area for the planimeter employed.
The solution of Eqs. (2), (7), and (8) then proceeds as follows:
0.3010
1.6684
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 163
colog (A + B) 8.0306
log (Z 2 + Zi) 8.9560n
colog (Zi  Zi) 0.3696n
log e sin co 9.3286
log (A  B) 9.7782n
log ^ ~ B b =\ cos « 7.8088n
log tan co 1.5198n
co 91?73
log sin co 9.9998
log e 9.3288
e 0.21
log (1 + e) 0.0828
log cos co 8.4800n
log K 1.6684
log cf ° 2312n
ordinate p — 1.7 km
.*. from curve t p = 98.4
T* = J. D. 2,416,457.75
log const. 4.1383
log it 1.6684
logP 2.0669
logVT^e" 9 " 02
log a sin i 7.8638
a sin i 73,000,000+ km
The preliminary values thus obtained are next tested by
comparing the velocities derived from them by Eqs. (10) with
the observed velocities. To illustrate, let us compute the
velocity for J. D. 2,416,496.0, twenty days after the origin
adopted in our curve. We have
t = 2,416,496.0 log cos (v + co) 9.8277n
t  T = +38.0 log K 1.6684
log (tT) 157978 1.4961n
log n 0.48942 g cos (v + co)  31.3 km
* T is here taken one revolution earlier than the date for the periastron
point marked on the curve. Using Eq. (8a) or (86) we obtain T = J. D.
2,416,458.0 which is adopted.
164 THE BINARY STARS
M 117?27 4Zj _ .3
v 136.01 V + 20.7
v + a, 227.74 * = 10.9 km
In this manner we compute as many velocities as necessary to
obtain a curve for comparison with the observed velocity curve.
In the present instance this was done for every tenth day, and
the results plotted as heavy black dots in the figure. By noting
the discrepancies, it appears that the branch on the apastron
side of the computed curve, if drawn, would be a little too
sloping, the other branch too steep, which indicates that the
computed value of e is a little too large. Changing this ele
ment and making the corresponding slight changes required
in T and o>, the test was repeated and after a few trials the
following elements were adopted as best representing the
observations :
V = +21.9 km
P = 116.65 days
e = 0.19
K = 46.5
w = 96?23
T = J. D. 2,416,459.0
a sin i = 73,000,000 km
The correction to the value of V was found last of all from
the residuals of the final ephemeris by the simple formula
[v]/n, where n is the number of observations and v the residual,
(o — c). The residuals from the final ephemeris and the final
curve may be found in Lick Observatory Bulletin, No. 122, 1907.
LehmannFilheV method may be termed the classical one,
and it is probably more generally used than any other. The
method proposed by Rambaut is considerably longer and
more involved than the later ones, and for that reason will not
be described here. Wilsing's method, as originally published,
was suitable only for orbits of small eccentricity, but Russell
later extended it to make it applicable to larger eccentricities
as well. This method is purely analytical, consisting in finding
a Fourier's series for the velocity in terms of the elements. It
should be very useful in special cases, particularly when the
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 165
period is so nearly a year that one part of the velocity curve
is not represented by any observations; but it is considerably
longer, in time consumed, than the method of LehmannFillwSs
and other geometrical methods to be described presently and
will not be further considered here.
Certain features of the methods proposed by Schwarzschild
and Zurhellen are both ingenious and practical. The following
account of them is taken in substance from Curtis' article
already referred to.
SCHWARZSCHUD'S METHOD
Given the velocity curve and the period, Schwarzschild first
determines the time of periastron passage. Let Mi and M 2 be
the observed velocities (i.e., the velocities measured from the
zeroaxis) of maximum and minimum, and draw the line whose
ordinate is (Mi + M 2 )/2. This line is the mean axis. Mark
upon it the points corresponding to P/2 and 3P/2; then lay a
piece of semitransparent paper over the plot, copy upon it the
curve together with the mean axis and mark also the points 0,
P/2, P, and 3P/2. Shift the copy bodily along the mean axis for
the 'distance P/2, and then rotate it 180° about this axis, i.e.,
turn the copy face downward on the original curve keeping
the mean axis in coincidence and bring the point or P of the
copy over the point P/2 of the original. The curves will then
cut each other in at least four points, and, in general, in four
points only. These will fall into two pairs, the points of each
pair separated by an abscissa interval P/2. The points of one
pair will be on different branches of the velocity curve, and it
is easy to see that, if v x and v 2 represent their true anomalies,
we shall have v 2 = v x + 180°. Now the only two points in the
true orbit which are separated by onehalf a revolution and for
which at the same time this relation of the true anomalies
holds are the points of periastron and apastron passage. Hence,
to select these points, choose the two points of intersection of
the curve and its copy which are separated by half a revolution
and which he on different branches of the curve. To distinguish
periastron from apastron we have the criteria: (1) at periastron
the velocity curve is steeper with respect to the axis than at
apastron; (2) the curve is for a shorter time on that side of the
mean axis on which the point of periastron lies.
166 THE BINARY STARS
This method is exceedingly good except when the eccen
tricity is small. In this case, co and T are quite indeterminate
and small errors in drawing the velocity curve will be very
troublesome. The method of LehmannFilh6s is then to be
preferred.
Having the value of T, the value of co is next found as follows:
From Eqs. (la) and (3) it is readily seen that the position of the
mean axis is
2 = V \ Ke cos co = Vi
and that, accordingly, the ordinate z' of any point measured
from the mean axis is
z' =j t  Vx = K cosO + «) (11)
Now at periastron v = 0°, at apastron v = 180°. Hence, if
we call the ordinates from the mean axis for these points z p '
and zj, we shall have
cos « = jjjr' or cos a = **' 2K* < 12 )
from which to determine w. This method is at its best when
co is near 90°.
Zurhellen has simplified Schwarzschild's method of finding
e, and we shall give this simpler form in connection with Zur
hellen's simple method of finding w.
ZURHELLEN'S METHODS
Zurhellen's method of determining co depends upon the rela
tions between the velocities for the two orbit points whose
true anomalies are ±90°. From Eq. (11) we have, when
v = 90°,
Zi = +K sin co
and when v = +90°,
Zi = —K sin co
Moreover, for these two points we have
Ei = — E2
M x = M a
(*i T) = (*, T)
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 167
Hence the two points are symmetrically placed with respect
to the mean axis in the Fcoordinate and with respect to the
point of periastron passage in the Xcoordinate. They may
therefore be determined by rotating the curve copy through
180° about the intersection of the ordinate of periastron with
the mean axis, and noting the two points of intersection of the
copy with the original curve. If the curve is prolonged through
one and 'onehalf revolutions, another point 180° from one of
these, say at +270°, can be determined in similar manner
and the location of all three can then be checked by drawing
the lines connecting the point v = +270° with v = —90°,
and v = —90° with v = +90°. These lines should cut the
mean axis at its intersections with the ordinates of periastron
and apastron, respectively. From the ordinates of the two
points v = ± 90°, measured from the mean axis, we have
sin co = — ,.„ ) or tan co = (12a)
2K z P — z a
from which to find co. The method is at its best when co is small.
Zurhellen's simplification of Schwarzschild's method of find
ing e is also based upon the relations between the two points
v = ±90°. Since
tan Y 2 E = tan x Av tan (45°  l A4>)
where <f> is the eccentric angle, we have, when v = + 90°,
E x = (90°  <6), E 2 = +(90°  c6)
Similarly,
sin c6 sin (90°  <f>)
M x = (90° c6) +
sin 1'
M, = +(90°  ♦)  ***<*£&*)
and therefore
*,_*,. w (h _ (i) . (180 . _ 2 „ _ smi^2*) (13)
The value of (fa — ii) may be read off directly from the dia
gram, and the value of (90° — 4>) can then be taken from the
table for Eq. (13), computed by Schwarzschild, which is given
168
THE BINARY STARS
below. Like the above method for finding o> this method is
best when co is small.
Schwarzschild's Table for the Equation
360°
2i\ — sin 2?7 = — — (< 2 — h)
ti — ti
ti — ti
<2 — ti.
V
P
V
P
V
P
0°
0.0000
30°
0.0290
60°
0.1956
1
0.0000
31
0.0318
61
0.2040
2
0.0000
32
0.0348
62
0.2125
3
0.0000
33
0.0380
63
0.2213
4
0.0001
34
0.0414
64
0.2303
5
0.0001
35
0.0450
65
0.2393
6
0.0002
36
0.0488
66
0.2485
7
0.0004
37
0.0527
67
0.2578
8
0.0006
38
0.0568
68
0.2673
9
0.0008
39
0.0611
69
0.2769
10
0.0011
40
0.0656
70
0.2867
11
0.0015
41
0.0703
71
0.2966
12
0.0020
42
0.0751
72
0.3065
13
0.0025
43
0.0802
73
0.3166
14
0.0031
44
0.0855
74
0.3268
15
0.0038
45
0.0910
75
0.3371
16
0.0046
46
0.0967
76
0.3475
17
0.0055
47
0.1025
77
0.3581
18
0.0065
48
0.1085
78
0.3687
19
0.0077
49
0.1147
79
0.3793
20
0.0089
50
0.1212
80
0.3900
21
0.0103
51
0.1278
81
0.4008
22
0.0117
52
0.1346
82
0.4117
23
0.0133
53
0.1416
83
0.4226
24
0.0151
54
0.1488
84
0.4335
25
0.0170
55
0.1561
85
0.4446
26
0.0191
56
0.1636
86
0.4557
27
0.0213
57
0.1713
87
0.4667
28
0.0237
58
0.1792
88
0.4778
29
0.0262
59
0.1873
89
0.4889
30
0.0290
60
0.1956
90
0.5000
Zurhellen also gives a method for finding the eccentricity
by drawing the tangents to the curve at the points of perias
tron. These can be drawn quite accurately except when the
periastron falls near a maximum or a minimum of the curve.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 169
Slight changes in its position will then introduce considerable
changes in the inclinations of the tangent lines.
The expression for the slope of a tangent may be written
dx _2ir dx _ 2t 1 dx
dt ~ P "dM ~ P" ' 1  e cos E ' dE
where x (= dz/dt) represents the ordinate drawn to the Faxis.
Also, by introducing the known values
cos E — sin <f> . cos <f> sin E
cos v = = j=r> sin v = 2 =
1 — e cos E 1 — e cos E
and transforming and simplifying we may write the fundamental
Eq. (3) in the form
dz rr . cos <f> cos w cos E — sin co sin E
x = r = K cos <t> • = ™
dt 1 — e cos E
Differentiating with respect to E, substituting and reducing,
we have
dx 27r„ , — cos <$> cos co sin E — sin co cos E + e sin co /1/IX
j7 = 5 K cos <£ r p^ (14)
dt P (1 — e cos Ey
At periastron E = 0° and at apastron E = 180°, whence we
have
dx —
2tK cos c/> sin
CO
dx
\2irK cos
4> sin co
dtp
P(l  e) 2
— >
dt a
P(l +
e) 2
and therefore
dx/dtp _
dx/dta
(l + <0 2
(1  eY
= "? 2
whence
f
,  9
 1
•,+1 (15)
KING'S METHOD
The methods of orbit computation so far described in this
chapter all rest upon the curve drawn to represent as closely
as possible the observed velocities and, at the same time, to
satisfy the conditions for elliptic motion. Unless the measures
are very precise, the first approximation will ordinarily not be
170 THE BINARY STARS
satisfactory. As stated on page 157, the only remedy is to
compute an ephemeris from the elements and, on the basis of
the residuals thus found, to draw a new curve. This process
is sometimes repeated three or four times before a curve is
found that will yield elements upon which a least squares
solution may be based.
The method devised by Dr. King, which is now to be pre
sented, aims to substitute a rapid graphical process for testing
/'
1\
^\
I
; S
^
k!
\
1 v<
A
Y
\
\
/
\
Fig. 11. — King's orbit method. Graph for e = 0.75, « = 60°.
the preliminary curve. Dr. King shows that a circle having
its center on the mean axis and a radius equal to K, the semi
amplitude of the velocity curve, "may be used as the equivalent
of the hodograph of observed velocities."*
Let the velocity curve and the circle be drawn (see Fig. 11)
and the abscissa distance corresponding to one revolution (P
being assumed to be known) be divided into any convenient
number of parts, say forty, f Now mark consecutive points
on the circumference of the circle by drawing lines parallel to
* For the proof of this relation the reader is referred to the original
article in Ap. Jour. 27, 125, 1908.
t An even number should be chosen, and it is obviously most convenient
to make the drawing upon coordinate paper.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 171
the mean axis at the intersections of the velocity curve with
the ordinates corresponding to successive values of the abscissa
and extending them to the circle. The circumference will
be divided into forty unequal parts, but these inequalities will
be found to vary uniformly. "The points will be close together
in the vicinity of one point of the circle, and will gradually
separate as we proceed in either direction therefrom, until
at the diametrically opposite point they reach their maximum
distance apart." These unequal arcs of the circle correspond
to the increase in the true anomalies in the orbit in the equal
time intervals, and therefore the point of widest separation
of the circle divisions corresponds to periastron, that of least
separation, to apastron. Further, the angle between the
Faxis and the periastronhalf of the diameter between these
two points is equal to w. To locate the point of periastron
on the velocity curve, find the intersection of the steeper branch
of the curve with a line drawn from the periastron point on the
circle parallel to the mean axis.
It is evident that the division points of the circumference
will be symmetrically disposed with respect to the apsidal
diameter (the diameter joining periastron and apastron points)
only when one of the division points in the line of abscissae
corresponds to an apse. In general, the periastron point will
lie within the longest division of the circumference, the apas
tron point within the shortest. If desired, the approximate
position of one of these points may be used as a new origin
from which to set off the fortieths of the period along the axis
of abscissae, and two division points on the circle may then be
brought into closer coincidence with the apsidal points.
Since dv/dt varies inversely as the square of the distance from
the focus, by measuring the lengths di and d 2 of the arcs at
points where v equals Vi and v%, we have
di _ (1 + e cos t>i) 2 ,
d 2 — (1 + e cos vz) 2
and hence if the arcs are measured at the points of periastron
and apastron where v equals 0° and 180°, respectively,
\/di — \/^2
d t \l  e) '
or e —
■\Zd~i + y/di
which determines e.
(16)
172 THE BINARY STARS
It is generally sufficiently accurate to measure the chords
instead of the arcs; when the eccentricity is high and the arcs
at periastron are inconveniently long, additional points of division
may readily be inserted.
It will be observed that this process furnishes a more thorough test
of the accuracy of the graph (velocity curve) than the method of equality
of areas. If it is imperfect, the points on the circumference of the circle
will not be distributed according to the regular order of increase or
decrease of the included arcs. If an ordinate of the graph is too long or
too short, the corresponding point on the circumference will be too near
or too far from the vertical diameter. If the points of maximum and
minimum velocity have not been well determined, the diameter of the
circle will be too long or too short. In the former case all the points
of the circumference will be crowded away from the vertical diameter; in
the latter, toward it.*
To test a given set of elements by comparison with the observa
tions proceed as follows:
Construct a circular protractor on some semitransparent
material (e.g., celluloid or linen tracing cloth) and divide it
into forty parts by radii to points on the circumference repre
senting the true anomalies for the given value of e correspond
ing to every 9° of mean anomaly (i.e., to fortieths of the period).
If the eccentricity is less than 0.77 the values of the true anomaly
can be taken directly from the Allegheny Tables.
On the plot of the orbit draw a circle of radius K with its
center on the mean axis and draw its vertical diameter. Set
the protractor upon the circle, making the centers coincide,
and turn the apsidal diameter of the protractor until it makes
an angle equal to w with the vertical diameter. Now note the
points where the radial lines representing the anomalies inter
sect the circumference of the circle. The abscissa axis of the
plot also having been divided into forty equal parts, erect
perpendiculars at the points of division equal to the corre
sponding ordinates of the circle. A freehand curve through the
extremities of these perpendiculars (i.e., ordinates to the mean
axis) gives the computed curve or ephemeris, and the residuals
can be read directly from the plot. The advantage of using
coordinate paper will be obvious.
From the account just given it will appear that King's method
is longer, or at least not shorter, than the others described
* King, he. cit.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 173
if only a single orbit is to be computed. But when orbit com
putation is to be taken up as a part of a regular program of
work, the method has very decided advantages. It is then
to be used as follows:
Let a set of protractors be constructed on transparent cellu
loid with radii representing the divisions in true anomaly for
every 9° of mean anomaly for the values e = 0.00, e = 0.05
to e = 0.95.
With the aid of these protractors draw curves on tracing
linen representing orbits with all values of e from 0.00 to 0.95
and all values of « from 0° to 360°. The intervals for e should
be 0.05, save for the larger values which are seldom used, and
for o), 15°. Practically, values of w to 90° will suffice, the curves
for the values in the remaining quadrants being obtained by
inverting the sheet and by looking through the linen from the
back in the two positions. Given the protractors, a complete
set of curves may be constructed in about 10 hours' time.
Having such a set of curves, plot the normal places for any
given binary star on the same scale as these curves in time and
in velocity.* Now place the standard curves upon the plot
until one is found that fits the observations. "If two or more
curves seem to give about equally good representations, it is
quite possible to interpolate elements between the graphs
plotted."f
By this process values of e correct to within one or two
hundredths and of w correct within a few degrees can generally
be obtained at the first trial and with an expenditure of less
than 10 minutes' time. The time of periastron passage fol
lows at once, and this set of preliminary elements may then
be used as the basis for a least squares adjustment. The pro
cedure has been found very satisfactory at the Dominion Observa
tory at Ottawa, and at the Dominion Astrophysical Observatory,
Victoria , B.C., where very many orbits of spectroscopic binary
stars have been computed.
* Since the velocity curve is ordinarily based on from fifteen to twenty
normal places the work of multiplying by the appropriate reduction factors
will require a very few minutes only. Of course, the amplitude of the curve
as well as the period must be known before the reduction factors can be
obtained. These are known with sufficient accuracy from the preliminary
plots.
tR. K. Young, Orbit of the Spectroscopic Binary 2 Sagittae, Jour.
R.A.S.C. 11, 127, 1917.
174
THE BINARY STARS
RUSSELL'S SHORT METHOD
Professor Henry Norris Russell has devised a graphical
method which is equally simple in its practical application.*
Write Eq. (la) in the form
p =V+^= V + Kecosa + Kcos (v + a) =
at
G + K cos (v + <o) (17)
where p represents the observed radial velocity.
Then (G + K) is the maximum, (G — K) the minimum
value of the velocity, so that G and K may be estimated at
once from the freehand curve. The period is also assumed
to be known. Equation (17) may then be written in the form
cos {V + Oj) = — ~ —
(18)
and the value of (v + a>) computed for each observed value of p.
If we subtract the corresponding values of M + Mo from each of
these, we shall have values of (v — M ) + (w — M ). The second part
of this expression is constant, while the first is the equation of the
center in the elliptic motion. During a revolution this varies between
equal positive and negative limits which depend only on the eccen
tricity, and are nearly proportioned to it, as is shown in the following
table.
Eccentricity
Maximum equa
tion of center.
0.10
11?5
0.20
23?0
0.30
34?8
0.40
46?8
0.50
59?2
0.60
72?3
0.70
86?4
0.80
102?3
0.90
122?2
If the values of (v — M) + (w — M ) are plotted against those of
M + M o, we obtain a diagram which, since it represents the relations
between the mean and the true anomalies, we may call the anomaly
diagram. If on this diagram a curve is drawn to represent the plotted
points, half the difference between its maximum and minimum ordi
nates will be the greatest value of the equatioD of the center, from which
e may be found at once by means of the table. The mean of the maxi
mum and minimum ordinates will be the value of w  M . The
instants when (v — M) f (co — Mo) has this value are those of periastron
and apastron passage, the former corresponding to the ascending branch
of the curve, which is always the steeper. The abscissae of the corre
Ap. Jour. 40, 282, 1914.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 175
sponding points of the curve are M and M + 180°. The values of e,
M , and co are now known, and the remaining elements may be found
at once from K and G.
„ According to Russell, the "principal advantage of this method
is that the form of the curves which give v — M as a function
of M depends upon e alone." For further details and an illus
trative example we refer the reader to the original memoir. Up
to the present time the method has not come into general use.
This remark applies also to the methods which have been
developed by Laves, Henroteau, and more recent writers on the
subject. References to the original papers for a number of these
methods are given at the end of this chapter.
DIFFERENTIAL CORRECTIONS TO THE ELEMENTS
Whatever method may be used in finding the preliminary
orbit it is generally advisable to determine the correction to the
elements by the methods of least squares. * The formula derived
by LehmannFilh6s from which the coefficients for the observa
tion equations are to be computed may be written as follows:
S. = dV + [cos (v + «) + e cos w]dK
at
, r S sin (v + a?) sin v, n , N 1 ,
+K\ cos co v 1 _ \ (2 + e cos v) \de
— i<L[sin (v + co) + e sin co]dco
* Publ. Allegheny Obs. 1, 33, 1908.
The advantages of applying the method of least squares to the definitive
solution of spectroscopic binary stars have been clearly stated by Schlesinger
in this paper. "The problem," he says, "involves the evaluation of five
quantities (six if the period be included) that are so interwoven as to make
their separate determination a matter of some difficulty. Herein lies the
chief advantage for this case of the method of least squares; for it enables
us to vary all of the unknowns simultaneously instead of one or two at a
time. . . . Further, the method of least squares not only has the advantage
of greater accuracy and of telling us how reliable our results are, but it
eliminates from the computations any personal bias or arbitrary step . . .
the method should be used in almost every case where the elements are not
avowedly provisional." Not all computers are so enthusiastic as to the
method. Judgment must of course be exercised in all orbit work as to
whether the data at hand warrant anything beyond the computation of
purely provisional elements. In spectroscopic binary orbits, for example,
such factors, among others, as the number and quality of the plates and their
distribution over the velocity curve, the character of the star's spectrum, and
the character of the orbit must be considered in making this decision.
176 THE BINARY STARS
+ sin (v + «)(1 + e cos y)* _^ ' e% y flT
 sin (v + «)(1 + e cos *>) 2 (1 5 e «) K« " ^ < 19 >
In practice the period is almost always assumed to be known
with accuracy and the last term of the equation is omitted.
To facilitate the computation, Schlesinger has transformed
this equation as follows: Put
a = 0.452 sin v(2 + e cos v)
_ (1 + e cos v) 2
* (1 + e) 2
T = dV + e cos w dK + K cos w de — Ke sin w dw
k = dK
T
= Kdu
ir 2  21 j
m = K X L^ • i dM, and w = (t> + «)
\1 — e 1 — e
Then the equations of condition take the form
(& = r + cos u • k + sin u • T + a sin u • e
+ p sin w • r + /3 sin w • (*  T)m (20)
The quantities a and /3 can be tabulated once for all and such
a tabulation is given by Schlesinger* so arranged "as to render
the normal equations homogeneous and to enable all multiplica
tions to be made with Crelle's tables without interpolation."
If this notation is used, the computer should have these tables
at hand.
When both spectra are visible on the plates, the orbits for
the two components with respect to the center of mass may be
determined separately. It is obvious that the two sets of
values of V, e, T, and P must be identical, the values of u> must
differ by 180°, while the two values for K depend upon the
relative masses of the components. The preliminary elements
* Loc. cU.
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 177
for the two components, when independently determined,
will, in general, not harmonize perfectly. To obtain the defini
tive values the best procedure is the one first suggested, I believe,
by Dr. King.* It consists in combining all the observations,
those for the secondary with those for the primary, into a single
set of observation equations (equations of condition) and solving
for one complete set of elements. If we write co' = (co + 180°)
and distinguish the values of K for the two components by
writing K x and K 2 , respectively, the equations in the notation
of Eq. (19) assume the form
d~jL = dV + [cos(t>+co) +■ e cos wJdKi+fcos^ + co')+e cosco'JciK^
at
, ([ sin (v + co) sin v (c . . N ~ v
+)\ cos co v ' 2 (2 + e cos i>) Ai
. T / sin (v + co') sin v, n . ,~\ v \ ,
+ cos co' l  e 2 ( 2 + e cos w ) \ K *\ de
— {[sin (v + co) + e sin co]Xi + [sin (v + co') + e sin co']X 2 }rfco
+ [sin (v + co)(l + e cos t>) 2 i£i
+ sin (v + co'Xl + e cos t>) 2 K 2 ] (1 * e ^ dT (21)
the value of the period being assumed to require no correction.
Since K 2 does not affect the residuals of the primary com
ponent, nor Ki those of the secondary, the terms containing
dK 2 and dK x disappear from the equations representing the
residuals from the primary and secondary curves, respectively.
SECONDARY OSCILLATIONS
When the orbit of a spectroscopic binary star has been com
puted and the theoretical velocity curve drawn, it is some
times found that the observed normal places are so distributed
with respect to the curve representing simple elliptic motion
as to suggest that a secondary oscillation is superimposed upon
it. The question is whether this grouping arises from some
source of error in the measurement of the spectrograms, from
* See Harper's paper, in Publ. Dominion Obs., 1, 327, 1914. Dr. Paddock
independently developed an equivalent equation. Lick Obs. Butt. 8, 166,
1915.
178 THE BINARY STARS
erroneous values of one or more of the orbit elements, or from
a real oscillation such as might be produced, for example, by
the presence of a third body in the system. This question has
been discussed by several investigators, among them Schlesinger,
Zurhellen, and Paddock. As early as 1911, Schlesinger, then
at the Allegheny Observatory, and his colleagues there, showed
that the "blend effect" caused by the overlapping of the absorp
tion lines of the two component spectra "may produce such an
apparent oscillation." They have also shown that it may be
produced by chance errors in the velocities derived from the
different lines of the spectrum, and they are convinced that a
critical analysis of the data will dispose of a considerable per
centage of cases wherein secondary oscillations have been
suspected. Later investigations by others have confirmed these
conclusions.
The possibility of a real secondary oscillation must, of course,
be recognized, and when a full analysis has shown that such an
oscillation is present, additional terms may be introduced into
the equations of condition to represent it upon the assumption
that it is produced by a third body revolving in a circular orbit
about one of the other two components. In the cases that have
arisen thus far this simple assumption has yielded a satisfactory
representation of the data, though it is apparent that there is no
reason for limiting such additional bodies to circular orbits.
Let T' represent the time when the secondary curve crosses
the primary from below, K' the semiamplitude of the sec
ondary oscillation, m' the ratio of the principal period to that
of the secondary oscillation, assumed to be known (it is gen
erally taken to be an integer), and put u' — m'n(t  T'),
t' = —m'uK'dT', k' — dK'; then the additional terms required
in Eq. (20) are
+ sin v! ■ k' + cos v! ■ r' (22)
For a more complete discussion of secondary oscillations the
reader is referred to the articles cited above.
ORBITS WITH SMALL ECCENTRICITY
In a circular orbit the elements T and « obviously have no
significance, and when the eccentricity is very small, they
become practically indeterminate by the geometrical or graph
ical methods which have here been described. Further, if
THE ORBIT OF A SPECTROSCOPIC BINARY STAR 179
approximate values are assumed, it is impossible to find cor
rections to both elements from the same least squares solution
because the coefficients for the differential corrections will be
nearly or quite equal. Some computers have overcome this
difficulty by assuming the preliminary value of co as final, and
deterinining corrections to T, but this is hardly a solution of
the problem. In such orbits the analytic method possesses
great advantages, as has been shown by several investigators,
notably Wilsing, Russell, Zurhellen, and Plummer. Paddock
has examined the question in great detail, extending some
of the earlier developments and adapting them for computation.
A full account of these methods would require more space than
is available here, and it has seemed best to refer the reader to
the original papers.
References
The following list of papers relating to one phase or another of the com
putation of orbits of spectroscopic binary stars, while not exhaustive, con
tains most of the more important ones.
Rambaut, A. A.: On the Determination of Double Star Orbits from Spectro
scopic Observations of the Velocity in the Line of Sight, Mon. Not.
R.A.S. 61, 316, 1891.
Wilsing, J.: tlber die Bestimmung von Bahnelementen enger Doppel
sterne aus spectroskopischen Messungen der GeschwindigkeitsCom
ponenten, A.N. 134, 89, 1893.
LehmannFilhbs, R.: tfber die Bestimmung einer Doppelsternbahn aus
spectroskopischen Messungen der im Visionsradius liegenden Ge
schwindigkeitsComponente, A.N. 136, 17, 1894.
Schwarzschild, K.: Ein Verfahren der Bahnbestimmung bei spectro
skopischen Doppelsternen, A.N. 152, 65, 1900.
Russell, H. N.: An Improved Method of Calculating the Orbit of a Spectro
scopic Binary, Ap. Jour. 16, 252, 1902.
: A Short Method for Determining the Orbit of a Spectroscopic
Binary, Ap. Jour. 40, 282, 1914.
Nltland, A. N.: Zur Bahnbestimmung von spektroskopischen Doppel
sternen, A.N. 161, 103, 1903.
Laves, K.: A Graphic Determination of the Elements of the Orbits of
Spectroscopic Binaries, Ap. Jour. 26, 164, 1907.
Zurhellen, W. : Der spectroskopische Doppelstern o Leonis, A.N. 173, 353,
1907.
: Bemerkungen zur Bahnbestimmung spectroskopischer Doppelsterne,
A.N. 176, 245, 1907.
: Weitere Bemerkungen zur Bahnbestimmung spectroskopischer
Doppelsterne, u.s.w., A.N. 177, 321, 1908.
t)ber sekondare Wellen in den GeschwindigkeitsKurven spectro
skopischer Doppelsterne, A.N. 187, 433, 1911.
180 THE BINARY STARS
King, W. F.: Determination of the Orbits of Spectroscopic Binaries, Ap.
Jour. 27, 125, 1908.
Curtis, H. D.: Methods of Determining the Orbits of Spectroscopic Bin
aries, Publ. A.S.P., 20, 133, 1908. (This paper has, with the author's
permission, been very freely used in preparing my chapter on the
subject.)
Plttmmer, H. C. : Notes on the Determination of the Orbits of Spectroscopic
Binaries, Ap. Jour. 28, 212, 1908.
Schlbsinger, F.: The Determination of the Orbit of a Spectroscopic
Binary by the Method of Least Squares, Publ. Allegheny Obs. 1, 33, 1908.
: On the Presence of a Secondary Oscillation in the Orbit of 30 H
Ursae Majoris, Publ. Allegheny Obs. 2, 139, 1911.
: A Criterion for Spectroscopic Binaries, etc., Ap. Jour. 41, 162, 1915.
Paddock, G. F.: Spectroscopic Orbit Formulae for Single and Double
Spectra and Small Eccentricity, Lick Obs. Bull. 8, 153, 1915.
Cttrtiss, R. H. : Method of Determining Elements of Spectroscopic Binaries,
Publ. Astron. Obs. Univ. of Michigan 2, 178, 1916.
Henroteatt, F. : Two Short Methods for Computing the Orbit of a Spectro
scopic Binary Star by Using the Allegheny Tables of Anomalies, Publ.
A.S.P. 29, 195, 1917.
King, E. S. : Standard Velocity Curves for Spectroscopic Binaries, Harvard
Ann. 81, 231, 1921.
Halm, J. K. E. : On a Graphical Determination of the Orbital Elements of a
Spectroscopic Binary, Mon. Not. R.A.S. 87, 628, 1927.
Picart, L. : B^emarques sur le calcul des orbites des 6toiles doubles spectro
scopiques, J.O. 10, 137, 1927.
Pogo, A. : On the Use of the Hodographic Method of Laves for Determining
Elements of Spectroscopic Orbits, Ap. Jour. 67, 262, 1928.
Maderni, A.: La determinazione degli elementi orbitali di una doppia
spettroscopica, Mem. Soc. Astron. Italiana N.S. 5, 65, 1930.
Orlopp, A. : Harmonic Tables for Spectroscopic Binaries, Odessa Astron. Obs.
1930.
Luyten, W. J.: A Rediscussion of the Orbits of 77 Spectroscopic Binaries (of
Small Eccentricities), Ap. Jour. 84 (1), 85103, July 1936.
In his Third Catalogue of Spectroscopic Binary Stars (Lick Obs. Bull. 11,
141, 1924) Dr. J. H. Moore lists, in separate tables, all stars for
which a variation in radial velocity had been fairly established,
before 1924.0. See also his Fourth Catalogue of Spectroscopic Binary
Stars (Lick Obs. Butt. 18, 138, #483, 1936).
CHAPTER VII
ECLIPSING BINARY STARS
We have seen that one of the first binary systems to be dis
covered with the spectrograph was Algol (0 Persei), long known
as a variable star. There are other stars whose light varies in
the same peculiar manner as that of Algol; that is, while it
remains sensibly constant at full brightness the greater part
of the time, at regular intervals it fades more or less rapidly
to a certain minimum. It may remain constant at this mini
mum for a short time and then recover full brightness, or the
change may be continuous. In either case the entire cycle of
change is completed in a small fraction of the time of constant
light between the successive niinima.
The hypothesis that in every such case the star, as viewed
from the Earth, undergoes a total, annular, or partial eclipse,
the eclipsing body being a relatively dark star revolving with
the other about a common center of gravity, completely accounts
for the observed facts and has been proved to be correct not
only in the one instance, Algol, but also in that of every Algol
type variable which has been investigated with the spectrograph.
Undoubtedly it is the correct explanation for all stars of this
type ; they are all binary systems.
Unless the darker star is absolutely nonluminous, there
should be a second ininimum when the bright star passes
between it and the Earth, the relative depth of the two minima
depending upon the relative intensity of the light of the two
stars and upon their relative areas. Such a secondary mini
mum has been observed in /3 Lyrae and in this star the light
is not quite constant at any phase, either maximum or min
imum. There is now no doubt but that this star and others
like it are also binary systems.
It was formerly thought that a distinction could be drawn
between variable stars of the type of Algol and those of the
type of j8 byrae, but measures with sensitive modern photom
eters, such as the selenium cell, the photoelectric cell, and
181
182 THE BINARY STARS
the slidingprism polarizing photometers, and measures of
extrafocal images on photographic plates have attained such
a degree of accuracy that a variation considerably less than
onetenth of a magnitude can be detected with certainty; and
it now appears that Algol itself not only has a slight secondary
minimum but that its light is nqt quite constant at maximum.
The distinction, therefore, breaks down and we may regard
all the stars of these two types as members of a single class,
calling them eclipsing binaries or eclipsing variables, according
to the point of view from which we take up their investigation.
In all, nearly 1,000 eclipsing binary stars are known at the
present time, a large percentage of them being too faint to
photograph with our present spectrographic equipment. It
is therefore a matter of great interest to inquire what informa
tion, if any, as to the orbits of these systems can be derived
from their light curves, the curves, that is, which are constructed
by taking the observed stellar magnitudes as ordinates and
the corresponding times as abscissae.
Up to the beginning of the present century observations of
eclipsing variables were made chiefly to determine the times of
minima accurately, with the object of improving the light ele
ments. As data accumulated, variations in the period began
to be noted in many instances. Dr. Seth Chandler in particular
was greatly interested in these variations and added parabolic
or periodic terms to the light elements in his catalogues of variable
stars published in the Astronomical Journal, but did not find an
explanation for them. For the most part, in fact, these varia
tions are so complicated that they are still a puzzle in celestial
mechanics, but in the case of Y Cygni* the variation has been
definitely traced to a revolution of the line of apsides, produced,
apparently, by the ellipsoidal figure of the component stars.
The earlier observations, confined as they were chiefly to
estimates of the time of minimum, were unsatisfactory material
for the determination of the orbital elements. Among the
pioneers in the accurate observation of the entire light curve
we may name Roberts and Nijland, using the method of estimates
developed by Argelander; Wendell and Dugan, using a polarizing
photometer devised by E. C. Pickering; Stebbins, using selenium
and photoelectric cells; and Parkhurst, Seares, and Baker, using
photographic methods.
♦Dugan, Contr. Princeton Univ. Obs. 12, 1931.
ECLIPSING BINARY STARS 183
Professor E. C. Pickering* made an investigation of the
orbit of Algol on the basis of its light curve as early as 1880,
and showed that a solution of such orbits was possible if certain
reasonable assumptions— for example, that the two stars are
spherical with uniformly illuminated disks and move in circular
orbits — were granted. The subject was resumed by him later,
and was taken up also by Harting, Tisserand, A. W. Roberts,
and others.
In the years 1912 to 1915, Russell and Shapley, at Princeton
University, f made a very thorough investigation of the problem,
Russell developing a general analytical method, which has
formed the basis of nearly all later calculations, and Shapley
applying it to the computations of the orbit elements of 90 sys
tems, 31 of the solutions being indicated as of the first grade.
Modifications of this method have been proposed by various
investigators, and Sitterly has developed a graphical method,
but in the present chapter Russell's method will be given.
In the most general case the problem is an extremely com
plicated one, for the orbits must be regarded as elliptical with
planes inclined at a greater or less angle to the line of sight,
and the two components as ellipsoids, the longest diameter of
each being directed toward the other star. Moreover, the
disks may or may not be uniformly illuminated; they may be
darker toward the limb, as our own Sun is, the degree of dark
ening depending upon the depth and the composition of the
enveloping atmosphere; and the side of each which receives
the radiation from the other may be brighter than the opposite
side. The complete specification of an eclipsing binary system
therefore requires a knowledge of at least 13 quantities which,
in Russell's notation, are as shown in the table on page 184.
The longitude of the node must remain unknown, as there is no hope of
telescopic separation of any eclipsing pair.
The value of a in absolute units can be found only from spectroscopic
data. In the absence of these, it is desirable to take a as an unknown
but definite unit of length, and express all other linear dimensions in
terms of it. Similarly, the absolute values of L x and L 2 can be deter
mined only if the parallax of the system is known. But in all cases the
* Dimensions of the Fixed Stars, with special reference to Binaries and
Variables of the Algol Type, Proc. Amer. Acad. Arts and Sciences 16, 1, 1881.
t References to their papers are given, with others, at the end of this
chapter.
184
THE BINARY STARS
combined light of the pair, L x + L 2 , can be taken as the unit of light
and the apparent brightness at any time expressed in terms of this.
This leaves the problem with eleven unknown quantities to be deter
mined from the photometric measures. Of these, the period is invari
ably known with a degree of accuracy greatly surpassing that attainable
for any of the other elements, and the epoch of principal minimum can
be determined, almost independently of the other elements, by inspec
tion of the lightcurve. Of the remaining elements, the constants
expressing ellipticity and reflection may be derived from the observed
brightness between eclipses. These effects are often so small as to be
detected only by the most refined observations. The question of
darkening toward the limb may well be set aside until the problem is
solved for the case of stars that appear as uniformly illuminated disks.
Orbital Elements
Semimajor axis
Eccentricity
Longitude of periastron
Inclination
Period
Epoch of principal conjunction
Eclipse Elements
Radius of larger star
Radius of smaller star
Light of larger star
Light of smaller star
and at least 3 constants defining
the amount of elongation, of
darkening at the limb, and of
brightening of one star by the
radiation of the other.
U
This leaves us with six unknowns. Fortunately, systems of such
short periods as those of the majority of eclipsing variables, usually have
nearly circular orbits (as is shown both by spectroscopic data and by the
position of the secondary minimum). The assumption of a circular
orbit is therefore usually a good approximation to the facts and often
requires no subsequent modification.
Russell's papers discuss first the simplified problem:
Two spherical stars, appearing as uniformly illuminated disks, and
revolving about their common center of gravity in circular orbits, mutually
eclipse one another. It is required to find the relative dimensions and
brightness of the two stars, and the inclination of the orbit, from the observed
light curve.
The determination of the orbit can be made by simple geometrical
methods, but their practical application demands the tabulation and
use of rather complicated functions.
We may assume P and t as already known. If the radius of the rela
tive orbit is taken as the unit of length, and the combined light of the
ECLIPSING BINARY STARS 185
two stars as the unit of light, we have to determine four unknown
quantities. Of the various possible sets of unknowns, we select the
following:
Radius of the larger star r i
Ratio of radii of the two stars *
Light of the larger star Li
Inclination of the orbit *
The radius of the smaller star is then r 2 = kr h and its light, L 2 = 1 — L\.
It should be noticed that, with the above definitions, k can never exceed
unity, but L 2 will exceed L x whenever the smaller star is the brighter
(which seems to be the fact in the majority of observed cases).
The development of the subject that follows is given, by his
courteous permission, largely in Russell's own words, taken
partly from his printed memoir but chiefly from a summary
sent me recently in manuscript form. The numbers for the
equations below are those given in his original paper.
The simplest case is that of a total eclipse which can often be recog
nized on inspection by a deep, fiatbottomed minimum in which the
light is Lx or X. Then L 2 = 1  X. If the loss of light at any moment
during the partial phase is ctL 2 the fraction a of the disk of the smaller
star must be obscured. By geometrical similitude a depends only on
the ratio k of the radii of the disks and the ratio 5/n of their projected
distance of centers to the larger radius, and we may write
«>(*■£)
The function / is transcendental but may be computed with ordinary
trigonometric tables. For any given value of k, we may invert this
function, and set
 = <K&, a) (9)
If is the longitude in the orbit, measured from mideclipse,
S 2 = cos 2 i + sin 2 i sin 2 = r^l/pffc, «)] 2 (11)
Now let a u a 2 , a% be any definite values of a and d u 2 , 03 the cor
responding values of (which may be found from the light curve).
Subtracting the corresponding equations of the form (11) in pairs, and
dividing one of the resulting equations by the other, we find
sin 2 0i  sin 2 2 \4>(k, «i)] 2  [<t>(k, <* 2 )] 2 _ ... v (12)
sin 2 2  sin 2 03 = [0(fc, « 2 )1 2  [<t>(k, «a)T 2 ~ ^' " h "* °* J K >
186 THE BINARY STARS
The first member of this equation contains only known quantities.
The second, if ct\, at, and a 3 are predetermined, is a function of k alone.
If this function is tabulated, the value of k in any given case can be
found by interpolation, or graphically. Equation (11) can then be
used to find n and i.
A theoretical light curve may then be found, which passes through
any three desired points on each branch of the observed curve (assumed
symmetrical). These points may be chosen at will by altering the
values of «i, a 2 , and a 3 . In practice it is convenient to keep a 2 and a 3
fixed, so that \f/ becomes a function of k and a\ only, and may be tabu
lated for suitable intervals in these two arguments. This has been
done in Table II, in which a 2 is taken as 0.6 and a 3 as 0.9. If A — sin 2 2 ,
B = sin 2 6 2 — sin 2 6 3 , (12) may be written
sin 2 0i = A + B^(k, ttl ) (13)
The points a and & on the light curve corresponding to a 2 and a 3 , together
with the point corresponding to any one of the tabular values of ai, then
give a determination of k. By taking a suitably weighted mean of
these values of k, a theoretical light curve can be obtained which passes
through the points a and 6, and as close as possible to the others. By
slight changes in the assumed positions of a and b (i.e., in the corre
sponding values of 6, or of t — t ), it is possible with little labor to obtain
a theoretical curve which fits the whole course of the observed curve
almost as well as one determined by least squares. The criterion of
this is that the parts of the observed curve below b (near totality),
between a and b, and above a (near the beginning or end of eclipse)
shall give the same mean value of k. The individual determinations
of k are of very different weight. Between a and 6 (that is for values of
a\ between 0.6 and 0.9) \f/ changes very slowly with k. At the beginning
and end of the eclipse the stellar magnitude changes very slowly with
the time, and hence, by (13), with yj/. The corresponding parts of the
curve are therefore ill adapted to determine k. For the first approxi
mation it is well to give the values of k derived from values of «i between
0.95 and 0.99, and between 0.4 and 0.2, double weight (provided the
corresponding parts of the curve are well fixed by observation). The
time of beginning or end of eclipse cannot be read with even approxi
mate accuracy from the observed curve and should not be used at all in
finding k. The beginning or end of totality may sometimes be deter
mined with fair precision, but does not deserve as much weight as the
neighboring points on the steep part of the curve. If further refinement
is desired, it can most easily be obtained by plotting the light curve for
two values of k and comparing with a plot of the observations. This
will rarely be necessary.
When once k is given, the determination of the light curve is a very
easy matter. For each tabular value of a t , Eq. (13) gives 6 U and hence
ECLIPSING BINARY STARS 187
(h  t ). The values of the stellar magnitude m corresponding to given
values of «i are already available, having been used in the previous work.
The light curve may thus be plotted by points in a few minutes.
After a satisfactory light curve has been computed, we may proceed
to determine the remaining elements. Let d' and 6" be the values
corresponding to the beginning of eclipse («i = 0) and to the beginning
of totality (ai = 1). Then by Eq. (13)
sin 2 d' = A + B+(k, 0) and sin 2 d" = A + B+(k, 1)
These computed values are more accurate than those estimated from
the freehand curve drawn to represent the observations. At the first
of these epochs 5 = n + r 2 , and at the second 5 = n  r 2 . We have
then, by Eq. (11)
ri 2(i + jfe)2 = cos 2 i + sin 2 i sin 2 d'
ri 2 (l  kY = cos 2 i + sin 2 i sin 2 d"
whence
4k cot 2 % = (1  fc) 2 sin 2 6'  (1+ *) a sin 8 0"
4Ati 2 (1 + cot 2 i) = sin 2 &  sin 2 6"
Introducing A and B, we have
4fc cot 2 i = MA + B[(l  *)V(*. 0)  (1 + *)V(*. 1)1
4AT! 2 cosec 2 i = B[^(k, 0)  M, 1)]
The coefficients are functions of k alone, and may be tabulated. It is
most convenient for this purpose to put the equations in the form
B
rS cosec 2 1 = ^ ^ ^
COt2l ' = *ffc) A
as in this way we obtain functions whose tabular differences are com
paratively smooth (which is not true of their reciprocals). With the
aid of these functions the elements may be found as soon as A and B
are known.
Occasionally cot 2 i comes out negative. The curve must then
be fitted as well as possible on the assumption of central transit
(cot i = 0).
The secondary eclipse will then be annular, with maximum
depth 1  X = fc 2 Li. When observed, it may be used to test
the applicability of the simplified theory. In a few cases the
188 THE BINARY STARS
eclipse at principal minimum may be annular. The process of
solution is identical, but the computed depth of secondary is
greater. If (1 — X) > k 2 only the total solution is possible,
since h\ cannot exceed 1. Otherwise the secondary minimum
must be observed to settle the question. When the eclipses
are partial (roundbottomed curves) an additional unknown
has to be found, «o, the maximum fraction of the area of the
smaller star which is eclipsed. The detailed analysis shows that
if only the primary minimum has been observed, a variety of
solutions, with different values of k and a , can be found, which
will give light curves practically indistinguishable even by the
best observations. The computed depth of secondary minimum,
however, differs from one solution to another ; and if this has been
observed, a definite determination is possible so long as the
primary is deep — or both eclipses fairly deep. When both
are shallow there is a considerable range of admissible solutions,
and a definite answer can be obtained only from additional
data, such as a spectrographs estimate of the relative brightness
of the components.
Russell's paper contains a number of tables, of which the
four directly applicable to the cases of total or annular eclipses
which have been discussed are reprinted here. His Table I,
tabulating the function given in Eq. (9), is omitted, though it is
fundamental, because it is used, so far as we are at present
concerned, only in constructing the subsequent tables.
His Table II contains the function ^(fc, a x ) defined by the equa
tion
ith ^  [1 + ^ (fc > ai)]2 ~ [1 + k P (k > " 2 > ]2
* K ' ai) " [1 + *p(*, « 2 )] 2  [1 + k V {k, a 3 )] 2
(where a 2 = 0.6 and a* = 0.9), which is used in determining k
in the case of total eclipse. The uncertainty of the tabular quan
tities does not exceed one or two units of the last decimal place,
except for the larger values of xf/, corresponding to values of a\
less than 0.3, for which the actual errors may be greater, but
are not more serious in proportion to the whole quantity
tabulated.
Table Ha contains the functions
<t>i(k) =
*(fc, 0)  *(fc, 1)
ECLIPSING BINARY STARS
189
o
TH
©
o
oo
H
P
OQ
CM
3
o
M
OQ
«
o
3
d
S§i83aisSS38S§a8388§agSS
^^^00 = 00 ooooooo
Jasgassselssssssssssssss
^ei ^ ^ ^ ^ ^ d d d d o o o o o o o o ^    ^
853883883888288858888838
C, N CH.HHHOOOOOOOOOOOHHHHH
sisaassssBiisissisaiSiM.
„ NN? HHHHOOOOOOOOOOOHHHHH
SS&S8S5l8SfcSSoS8^S5S5Ssi:Sa5
^ei««HHHOJed ? OOOOOOHHHHH
iS3B88S3g»88S§S88iaiS«B?
^•««HHHHOOOOooeeeoHHHHH
5§S8S§I3S5i3a§Si8IS§aS8S
4. j + + + I I I • I I ' ' ' '
»^« N ciHHHOOOOOOOOOOHHHHH
aBS8aaaa»8ES88asaBB83E8a
553B83S833S38S833M.383M
_ — WMQ
3gS8S8SgS3SSS8S5sS88S88S
dx^g^gc^^dooooooo^^^
QO>00!0
I©
IOQOAO
ddddddddddddooooooooooo^H
8§§2^§SSS§^^SSSoS?:S5««*o5«
190
and
<t>*(k) =
THE BINARY STARS
4&
(l  *)V(*, 0)  (l + *)V(*, i)
which are useful in determining the elements in the case of total
eclipse.
Table Ila. — For Computing the Elements in the Case op Total
Eclipse
k
*i(*0
fa(k)
1.00
0.380
0.939
0.95
0.401
0.894
0.90
0.417
0.848
0.85
0.427
0.802
0.80
0.431
0.755
0.75
0.431
0.709
0.70
0.427
0.663
0.65
0.419
0.617
0.60
0.406
0.572
0.55
0.390
0.527
0.50
0.371
0.482
0.45
0.349
0.436
0.40
0.323
0.390
0.35
0.294
0.345
0.30
0.262
0.298
0.25
0.226
0.250
0.20
0.187
0.202
0.15
0.145
0.153
0.10
0.100
0.103
0.05
0.052
0.052
0.00
0.000
0.000
Table A gives the loss of light (1 — X), corresponding to a
given change Aw in stellar magnitude. For a difference of
magnitude greater than 2.5, the loss of light is 0.9000 + one
tenth of the tabular value for Am — 2™5. Table B gives the
values of (0 — sin 0) for every 0.01 of (expressed in circular
measure), and saves much labor in computing the values of sin
corresponding to a given interval for minimum.
ECLIPSING BINARY STARS
191
Table A.— Loss of Light Corresponding to an Incbease Am in Stellab
Magnitude
o.o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
0.0000
0.0880
0.1682
0.2414
0.3082
0.3690
0.4246
0.4752
0.5214
0.5635
0.6019
0.6369
0.6689
0.6980
0.7246
0.7488
0.7709
0.7911
0.8095
0.8262
0.8415
0.8555
0.8682
0.8798
0.8904
0.0092
0.0964
0.1759
0.2484
0.3145
0.3748
0.4298
0.4800
0.5258
0.5675
0.6055
0.6403
0.6719
0.0183
0.1046
0.1834
0.2553
0.3208
0.3806
0.4351
0.4848
0.5301
0.5715
0.6092
0.6435
0.6749
0.7008 0.7035
0.72710.7296 0.7321
0.0273
0.1128
0.1909
0.2621
0.3270
0.3862
0.4402
0.4895
0.5344
0.0362
0.1210
0.1983
0.2689
0.3332
0.0450
0.1290
0.2057
0.2756
0.3393
0.3919 0.3974
0.4454 0.4505
0.4942
0.5387
0.5754 0.5793
0.4988
0.6127
0.6468
0.6779
0.7062
0.7511
0.7730
0.7930
0.8112
0.8278
0.8430
. 8568
0.8694
0.8809
0.8914
0.9000 0.9009
0.7534
0.7751
0.7949
0.8129
0.8294
0.8444
0.8581
0.8706
0.8820
0.8924
0.7557
0.7772
0.7968
0.8146
0.8310
0.0538
0.1370
0.2130
0.2822
0.3454
0.4030
0.4555
0.5034
0.5429 0.5471
0.5831 0.5870
0.6163
0.6501
0.6808
0.7089
0.7345
0.7579
0.7792
0.7986
0.8163
0.8325
0.6198
0.6533
0.6838
0.7116
0.7370 0.7394 0.7418
0.8458 0.8472
0.8594
0.8718
0.8831
0.8933
0.9018 0.9027
0.8607
0.8729
0.8841
0.8943
8 9
0.6233
0.6564
0.6867
0.7142
0.0624
0.1449
0.2202
0.2888
0.3514
0.4084
0.4605
0.5080
0.5513
0.5907
0.6267
0.6596
0.6895
0.7169
0.7601
0.7812
0.8005
0.8180
0.8340
0.8486
0.8620
0.8741
0.8852
0.8953
0.7623
0.7832
0.8023
0.8197
0.8356
0.8500
0.8632
0.8753
0.8862
0.8962
0.9036 0.9045 0.9054
0.0710
0.1528
0.2273
0.2953
0.3573
0.4139
0.4654
0.5125
0.5554
0.5945
0.6302
0.6627
0.6924
0.7195
0.7441
0.0795
0.1605
0.2344
0.3018
0.3632
0.4192
0.4703
0.5169
0.5594
0.5982
0.6336
0.6658
0.6952
0.7220
0.7465
0.7645
0.7852
0.8041
0.8214
0.8371
0.8514
0.8645
0.8764
0.8873
0.8972
0.9062
0.7667
0.7872
0.8059
0.8230
0.8386
0.8528
0.8657
0.8775
0.8883
0.8981
0.9071
0.7688
0.7891
0.8077
0.8246
0.8400
0.8541
0.8670
0.8787
0.8893
0.8991
0.9080
For values of Am greater than 2.5, the loss of light is 0.9000 plus Ho of the loss of light
corresponding to Am — 2.5.
Table B. — Values of — sin
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.00
0.0000
0.0002
0.0013
0.0045
0.0105
0.0206
0.0354
0.0558
0.0826
0.1167
0.01
0.0000
0.0002
0.0015
0.0049
0.0114
0.0218
0.0372
0.0582
0.0857
0.1205
0.02
0.0000
0.0003
0.0018
0.0055
0.0122
0.0231
0.0390
0.0607
0.0889
0.1243
0.03
0.0000
0.0004
0.0020
0.0060
0.0131
0.0244
0.0409
0.0632
0.0920
0.1283
0.04
0.0000
0.0005
0.0023
0.0066
0.0141
0.0258
0.0428
0.0658
0.0953
0.1324
0.05
0.0000
0.0006
0.0026
0.0071
0.0151
0.0273
0.0448
0.0684
0.0987
0.1365
0.06
0.0000
0.0007
0.0029
0.0078
0.0161
0.0288
0.0469
0.0711
0.1022
0.1407
0.07
0.0001
0.0008
0.0033
0.0084
0.0171
0.0304
0.0490
0.0739
0.1057
0.1450
0.08
0.0001
0.0010
0.0037
0.0091
0.0183
0.0320 0.0512
0.0767
0.1093
0.1494
0.09
0.0001
0.0011
0.0041
0.0098
0.0194 0.0337 JO. 0535
0.0796
0.1130
0.1539
192
THE BINARY STARS
To illustrate Russell's method I have chosen his orbit of
W Delphini, which is a "typical Algol variable with a deep
primary minimum, showing a constant phase, and little or no
secondary minimum." Its light curve, "denned by the 500
observations by Professor Wendell, with a polarizing photom
eter, which are published in the Harvard Annals, 69, Part 1,"
Table o. — Observed Magnitudes
Phase
Mag.
No.
obs.
O C
Phase
Mag.
No.
obs.
O C
0^2894
9.41
6
+0"?01
+0^0560
11.76
7
+0"?01
0.2637
9.49
5
+0.02
0.0659
11.58
8
+0.01
0.2458
9.58
5
+0.04
0.0753
11.33
7
0 .04
0.2306
9.59
4
0.01
0.0859
11.14
5
0.02
0.2200
9.67
5
0.00
0.0937
10.97
5
0.05
0.2106
9.73
8
+0.01
0.1036
10.88
8
+0.02
0.2007
9.79
10
0.00
0.1147
10.73
8
+0.05
0.1911
9.88
12
+0.02
0.1246
10.56
12
+0.03
0.1817
9.95
10
+0 .01
0.1351
10.39
14
0.00
0.1718
10.02
8
.00
0.1445
10.31
11
+0.04
0.1615
10.16
17
+0.04
0.1546
10.13
10
0.02
0.1506
10.23
14
0.00
0.1641
10.10
11
+0.04
0.1396
10.37
14
+0.01
0.1744
9.97
10
0.00
0.1311
10.44
16
0.03
0.1847
9.90
9
+0.02
0.1212
10.59
17
0.03
0.1941
9.79
9
0.02
0.1121
10.78
14
+0.01
0.2050
9.71
8
0.02
0.1013
10.91
17
0.04
0.2157
9.71
6
+0 .04
0.0906
11.12
14
0.01
0.2242
9.63
8
+0.01
0.0809
11.30
10
0 .02
0.2345
9.57
7
0.00
0.0715
11.51
12
0.00
0.2507
9.50
7
0.00
0.0617
11.69
10
.00
0.2708
9.48
7
+0.03
0.0509
11.88
7
0.00
0.2811
9.43
4
+0.02
0.0313
12.05
5
0 .04
0.94
9.42
5
+0.02
0.0169
12.08
4
0 .02
1.90
9.35
5
0.05
0.0082
12.07
7
0.03
2.04
9.41
7
+0.01
+0.0060
12.16
5
+0.06
2.67
9.38
5
0.02
0.0139
12.09
4
0 .01
3.04
9.42
3
+0.02
0.0261
12.03
5
0.07
4.04
9.44
6
+0 .04
0.0356
12.02
6
0 .03
4.48
9.36
7
0.04
+0.0460
11.87
6
0.04
is shown in Fig. 12. The observations have been combined
into the normal places given in Table o, on the basis of a period
of 4.8061 days, which was found to require no correction.
ECLIPSING BINARY STARS
193
From the 38 observations outside minimum we find the magnitude
during constant light to be 9^395 + 0.009. There is no evidence of
any change during this time. With a circular orbit, the secondary
minimum should occur at phase 2?40. As none of the observations fall
within 0?27 of this, they give us no information whether such a minimum
exists. The light curve of the principal minimum is very well deter
mined. The eclipse lasts from about 0?28 to +0?28, and there is a
short constant period at the middle, of apparently a little less than one
tenth the total duration of the eclipse. The mean of the 20 observa
tions lying within 002 of the middle of eclipse gives for the magnitude
at this phase 12^10 ± 0.014. The range of variation is therefore 2?70,
9^0
lo'no
llTO
12T»0
0$3 "0^2 OSll ($0 +0?1 +0?2 +0?3
Fig. 12. — Light curve of the principal minimum of W Delphini.
and the light intensity at minimum 0.0832 times that at maximum.
This shows at once that the eclipse is total, for if it was annular, the
companion (even if perfectly dark) must cut off at least 0.917 of the
light of the primary, and hence its radius cannot be less than 0.956 times
that of the latter. In such a system the duration of the annular phase
could not exceed 0.044/1.956, or 0.022 of the whole duration of eclipse.
The observed constant phase is almost five times as long as this.
The brighter star, therefore, gives 0.9168 of the whole light of the
system, and if isolated would appear of magnitude 9.49; while the
fainter but larger star which eclipses it gives out only oneeleventh as
much light, and when seen alone at minimum is of magnitude 12.10.
The loss of light (1  I) at any given time, t, will be 0.9168a!,
since a x is the percentage of obscuration. For a series of values
of a x we tabulate the values of (1  I) and then take from
Table A the corresponding changes of magnitude and apply
them to magnitude 9.4. Next, from the freehand curve drawn
194
THE BINARY STARS
to represent the data of observation, we read off the epochs
h and U at which the magnitudes so computed are reached
before and after the middle of eclipse. Half the difference
of t\ and t 2 may be taken as the interval t from the middle of
eclipse to each phase and the corresponding value of 6 formed
from
= pt = 1.3065J, where is expressed in radians and t in days.
With the aid of Table B sin 6 is found and then sin 2 0.
quantities are all entered in Table b.
Table 6
These
a\
1  I
Mag.
d
h
sin*
sin 2  A
* (A, ai)
k
0.0
0.0000
9°400
04304:
+04300:
0.394:
0.1474:
0.1105:
+4.28:
0.56:
0.1
0.0917
9 .505
0.2540
0.2515
0.3304
0.1050
0.0681
2.64
0.504
0.2
0.1834
9 .620
0.2286
0.2258
0.2968
0.0860
0.0491
1.908
0.505
0.3
0.2760
9.749
0.2075
0.2030
0.2681
0.0702
0.0333
1.290
0.480
0.4
0.3667
9.896
0.1884
0.1830
0.2426
0.0576
0.0207
0.802
0.462
0.5
0.4584
10 .066
0.1682
0.1644
0.2173
0.0462
0.0093
+0.361
0.36:
0.6
0.5500
0.6417
10 .266
10 .514
0.1486
0.1270
0.1470
0.1274
0.1931
0.1661
0.0369
0.0272
0.0000
0.0097
0.7
0.376
0.64:
0.8
0.7334
10 .835
0.1070
0.1048
0.1381
0.0190
0.0179
0.694
0.56:
0.9
0.8250
11 .292
0.0824
0.0788
0.1054
0.0111
0.0258
1.000
0.9S
0.8709
11 .624
0.0655
0.0624
0.0886
0.0071
0.0298
1.156
0.58:
0.98
0.8986
11 .884
0.0505
0.0462
0.0632
0.0040
0.0329
1.277
0.525
0.99
0.9076
11 .985
0.0430
0.0390
0.0536
0.0029
0.0340
1.318
0.528
1.00
0.9168
12 .100
0.021:
+0.019:
0.026:
0.0007:
0.0362:
1.404:
0.50:
1 Russell's computations were made with a slide rule. Repeating them with fiveplace
logarithms, I obtain figures, which sometimes differ slightly from those tabulated. These
differences, however, are unimportant for they produce no appreciable changes in the final
elements.
From the values of h and t% it appears that the observed curve is
remarkably symmetrical, and that the actual epoch of mideclipse is
0.0015 days earlier than that assumed by Wendell. The times of
beginning and ending of the eclipse cannot be read accurately from
the curve and are marked with colons to denote uncertainty.
From the values of sin 2 we have now to find k with the
aid of Table II. From Eq. (13) we have
<Kfc, «i) =
sin 2 di  A
B '
hence, if we let A be the value of sin 2 when a x = 0.6 and A — B
its value when ct x — 0.9, we may find a value of k for every
ECLIPSING BINARY STARS
195
tabulated value of «i by inverse interpolation in Table II.
Thus, taking A = 0.0369 and B(= sin 2 2  sin 2 3 ) = 0.0258,
as given by our curve, we find for ai = 0.0 that
*(fe,«i) = +4.28:
and hence, from the first line of Table II, k = 0.56:. Colons
are here used because the values of k are less accurate when
the tabular differences of f(k, oi) are small.
The values of k are seen to be fairly accordant except for
those corresponding to values a x near 0.6. Inspection of Table II
"shows that this discrepancy may be almost removed by increas
ing all the values of + by 0.024 which may be done by dimin
ishing A by 0.0245. Our new value of A is therefore 0.0363."
The new set of fc's are found to be discordant for values of «i near
0.9; "but by diminishing B by 2.5 per cent [giving B = 0.0252]
and hence increasing all the computed values of * in the corre
sponding ratio, we obtain a third approximation of a very satis
factory character." The general mean is now k = 0.528.
With these final constants, A = 0.0363, B = 0.0252, k = 0.528,
we may compute a theoretical light curve and also the elements
of the system from Eq. (14). Table c gives the second and
third approximations to the value of k and the data for the
final light curve.
Table c
2d Approx.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.98
0.99
1.00
+4.30:
2.665
1.932
1.314
0.826
0.385
+0.024
0.352
0.670
0.976
1.131
1.253
1.294
1.38:
0.56:
0.512
0.517
0.500
0.503
0.47:
0.48:
0.40:
0.714
0.610
0.597
0.55:
3d Approx.
+ 4.40:
2.73
1.974
1.344
0.845
0.394
+0.025
0.360
0.685
1.000
1.157
1.282
1.324
1.412
0.58:
0.534
0.538
0.527
0.532
0.51:
0.54:
0.51:
0.564
0.507
0.512
0.48:
+4.100
2.713
1.949
1.348
0.843
+0.400
0.000
0.358
— 0.689
1.000
1.162
1.276
1.318
1.389
Final light curve
^(0.528, ai)\ B+
0.1032
0.0683
0.0491
0.0338
0.0212
0.0101
0.0000
0.0090
0.0173
0.0252
0.0293
0.0322
0.0332
0.0350
sin"
0.1395
0.1046
0.0854
0.0701
0.0575
0.0464
0.0363
0.0273
0.0190
0.0111
0.0070
0.0041
0.0031
0.0013
sin $
(
0.373
0.292
0.324
0.252
0.292
0.227
0.265
0.205
0.240
0.185
0.215
0.166
0.191
0.146
0.165
0.127
0.138
0.106
0.105
0.081
0.084
0.064
0.064
0.049
0.056
0.043
0.086
0.028
196 THE BINARY STARS
Plotting the magnitudes computed in Table a against the epochs
—0.0015 ± t, we obtain the computed light curve. The residuals
(O — C) are given in the last column of Table a. Their average value,
regardless of sign, is 0?020.
From Table Ha we find for k = 0.528, $ x (ib) = 0.382,
<h(k) = 0.507;
whence
cot 2 % = ^jjT  A = 0.0133, cot i = 0.115, i = 83° 25'
n* cosec 2 % = rjyr = 0.0660, ri 2 = 0.0652, n = 0.256
<Pi(k)
and finally
r 2 = kri = 0.135.
In other words, we have, taking the radius of the orbit as unity,
Radius of larger star . 256
Radius of smaller star . 135
Inclination of orbit plane 83° 25'
Least apparent distance of centers 0.114
Light of larger star . 0832
Light of smaller star . 9168
Period of revolution 4 . 8061 days
At the middle of eclipse, the larger star overlaps the other by only
0.007 of the radius of the orbit, or about onetwentieth of the radius
of the smaller body, so that the eclipse is very nearly grazing. The
smaller star gives off eleven times as much light as the other, and exceeds
it fortyfold in surface brightness.
The loss of light at secondary minimum should be k times the light
of the fainter star, or 0.023 of that of the system. The corresponding
change in stellar magnitude is 0.027, which could only be detected by
refined observations.
For disks darkened toward the limb the solution proceeds
along essentially similar lines. The function / must be deter
mined by numerical integration. The principal difference is
that an annular eclipse is not flatbottomed, since more light is
cut off when the companion obscures the bright center than
when it obscures a region near the limb. (An actual case of this
sort has been reported by McDiarmid, TX Cassiopeiae.*)
* Contr. Princeton Univ. Obs. No. 7, 1924.
ECLIPSING BINARY STARS 197
The changes of light at the beginning and end of eclipse are slower,
so that the times of first and last contact, calculated from the
deeper parts of the eclipse tend to be farther apart, and the
computed diameter of the brighter star, at least, larger than for
the "uniform" solution.
Tables for disks in which the brightness falls off to zero at the
limb have been computed by Russell and Shapley.*
Orbital eccentricity produces no perceptible asymmetry in
the form of the light curve for a single minimum but displaces
the secondary minimum from the halfway point between the
O
Sun
C^^_
€>
Fig 13 —The system of W Delphini. Two relative orbits of the bright star
are shown, the upper one representing the elements as given in the accompanying
solution, the lower, Shapley's, on the assumption that the stars are darkened to
zero at the limb. The diameters of the disks of stars and Sun are drawn on the
same scale. The three bodies are of equal mass, but the stars are less dense than
the Sun. {From Shapley's artide in Povular Astronomy, 20, 572, 1912.)
primary minima, and may change the durations and depths of
both. It is even theoretically possible for the eclipse to fail
altogether near apastron. The component e cos w which dis
places the epoch of secondary, can often be very accurately
determined from the observations; the component e sin «, which
alters the length of minimum, is very hard to find with accuracy.
In one noteworthy case, Y Cygni, to which reference has
been made on an earlier pagef a rotation of the line of apsides
detected photometrically has been confirmed by spectrographs
observations.
Ellipticity of the components, caused by their mutual attrac
tion, has been found in practically all cases where pairs of small
separation have been well observed. It causes the light curve
* Ap. Jour. 36, 239, 385, 1912.
t See p. 182.
198 THE BINARY STARS
to be bowed up between the minima, with maxima halfway
between them. Its amount increases as the relative distance
separating the two stars diminishes, in close agreement with
Darwin's theoretical calculation for masses of homogeneous
fluid. The computed ellipticity of figure corresponding to a
given light curve depends, however, on the assumed degree of
darkening at the limb, so that it is the relative, rather than the
absolute, values which are of importance.
For certain systems whose components are almost in contact
(VW Cephei, etc.) the greater part of the variation is due to
ellipticity of figure.
The "reflection" effect* arises from the heating of the com
panion by the radiation of the primary, so that the side turned
toward the latter is brighter than the opposite side. In con
sequence, the light curve is higher just outside the secondary
minimum than outside the primary. This effect is small com
pared with the total light but may amount to a very large part
of the light of the secondary. In Y Camelopardalis, indeed, the
side of the secondary which is turned away from the primary
appears to be almost completely dark.
The information obtained from eclipsing variables is of far
reaching value. When supported by spectrographic observa
tions, it provides the most complete knowledge of a stellar
system that we can hope for at present — masses, linear diameters,
densities, surface brightness (when the parallax is known, as for
the distant companion of Castor), and even something regard
ing the law of internal density from the motion of periastron
(Y Cygni). Even without this potent aid we may obtain
data otherwise inaccessible regarding the relation of surface
brightness to spectral type and color, and especially concerning
stellar densities.
The equations for computing the latter are simple. Let the
total mass of the system be m, that of the larger star my, and
that of the smaller m(l — y). If a is the semimajor axis of the
orbit, we shall have a = Km V3 P % , where £ is a constant
depending on the units of measurement. If we choose the
Sun's mass, the Sun's radius, and the day as units, then, for
the Earth's orbital motion, a = 214.9, P = 365.24, whence
K = 4.206.
* First detected by Dugan in the light curve of RT Persei and by Steb
bins in that of Algol.
ECLIPSING BINARY STARS 199
In determining the elements of the system we have taken a
as our unit of length. The actual radius of the larger star is
therefore ari, and its volume, in terms of the volume of the
Sun, if 3 mP 2 ri 3 , or 74.4raPVi 3 . Hence its density is
= 0.01344?/
and similarly that of the smaller star,
(i y)
p 2 = 0.01344
PW
If the magnitudes and spectral classes are known, the ratio
y/(l — y) can be estimated closely with the aid of the mass
luminosity law. This procedure will give close approximations
to the true densities, particularly for the brighter and more
massive components.
Russell summarizes the conclusions he has drawn from his
extensive statistical investigations as follows:
The great majority of them (the brighter components) belong
to the main sequence, and are of classes B8 to A5. These
are strikingly similar in density, clustering closely about a mean
value 0.32 times that of the Sun. (This is the geometric mean
which in a case like this is more representative than the arith
metic.) Every spectral type is, however, represented, the whole
main sequence, from the Mdwarf YY Geminorum to the super
giant Otype stars Y Cygni* and H.D. 1337, f Giants are fewer,
but cover a wide range of spectral type and density. Only the
white dwarfs are absent. (It is worth notice in passing that a
pair of white dwarfs revolving nearly in contact might have a
period of only two or three minutes !)
The faint components of eclipsing systems are usually larger
than their primaries. This is a conspicuous example of observa
tional selection — small companions producing only shallow
eclipses which are unlikely to be discovered. The stars thus
selected appear to belong to a rather unusual type, intermediate
between ordinary giants and dwarfs, about which we would
otherwise know next to nothing.
In Table 1, I have listed the dimensions, masses and densities
of 22 systems for which complete spectrographic and photometric
* Princeton Contrib. No. 12, 1931.
t Publ. Dom. Ap. Obs. 3, 275, 1926.
200
THE BINARY STARS
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ECLIPSING BINARY STARS 201
data are available, taking the figures from the authorities cited.
The stars are arranged in the order of spectral class of the
primary, and the various columns give, in addition to the star's
name, spectral class and approximate revolution period, the
radii of the two components (6, bright, /, faint), their masses,
and their densities, all in terms of the Sun as unit, and the
distance between their centers in millions of kilometers.
There are many additional systems for which similar data
have been or might be computed with the aid of the mass lumi
nosity relation, on the basis of the light curves supplemented, in
some instances, by spectrographic orbits of the brighter com
ponent. In fact, in his recent memoir Die Bedeckungsverdnder
lichen," Gaposchkin gives such data for 218 of the 349 eclipsing
variables (many of them as faint as the tenth magnitude) he
catalogues. In many cases his figures are but rough approxima
tions, computed, without even the aid of a light curve, from the
observed depths of the minima at the two eclipses, and the
observed durations of the eclipses. For 54 systems he gives
more precise values for some or all of these quantities.
Utilizing all of the available data, Gaposchkin discusses not
only the complicated relations between mass, density, lumi
nosity, and spectral type in the eclipsing variables but also the
general characteristics of their orbits and the question of their
galactic concentration. He finds, among other results, that
there is a definite concentration toward the plane of the Milky
Way, most markedly in the case of the pairs of spectral classes
O and B, and that the eccentricity of the orbit increases both
with increasing period and with the spectral class as we pass
from A toward M.
References
Pickering, E. C. : Dimensions of the Fixed Stars with Special Reference to
Binaries and Variables of the Algol Type, Proc. Amer. Acad. Arts and
Sciences 16, 1, 1881.
Roberts, A. W. : On the Relation Existing between the Light Changes and
the Orbital Elements of a Close Binary System, Mon. Not. R.A.S.
63, 527, 1903.
: On a Method of Determining the Absolute Dimensions of an Algol
Variable Star, Mon. Not. R.A.S. 66, 123, 1906.
Russell, H. N. : On the Determination of the Orbital Elements of Eclipsing
Variable Stars, Ap. Jour. 35, 315, 1911; 36, 54, 1912.
Russell, H. N., and H. Shapley: On Darkening at the Limb in Eclipsing
Variables, Ap. Jour. 36, 239, 385, 1912.
202 THE BINARY STARS
Shapley, H. : The Orbits of Eightyseven Eclipsing Binaries — A Summary,
Ap. Jour. 38, 158, 1913.
: A Study of Eclipsing Binaries, Contrib. Princeton Univ. Obs. 3, 1915.
Vogt, H.: Zur Theorie der Algolveranderlichen, Veroff. Sternwarte Heidel
berg 7, 183, 1919.
Fetlaar, J.: A Contribution to the Theory of Eclipsing Binaries.
Recherches Astron. de VObs. d' Utrecht 9, Pt. 1, 1923.
Sitterly, B. W.: A Graphical Method for Obtaining the Elements of
Eclipsing Variables, Pop. Astron. 32, 231, 1924; Contrib. Princeton Univ.
Obs., #11, Pt. I, 1930.
Scharbe, S. : Bestimmung der Kreisbahnen der Veranderlichen vom Algol
typus nach der Helligkeitskurve, BuU. Obs. Central de Russie 10, No. 94,
1925.
Gaposchkin, S.: Die Bedeckungsveranderlichen, Veroff. Univ. Stern
wartezn BerlinBabelsberg 9, No. 5, 1932.
W. Kbat: Some Remarks on the Determination of the Orbital Elements of
Eclipsing Variable Stars, Veranderliche Sterne, Forschungs und
Informations Bulletin 4, 97, 1933. NishniNovgorod. The note
presents a generalization of Russell's method extending it to apply
also to systems of great eccentricity. The formulas for finding e and a
from the light curve are free from the assumption that i = 90°.
Walter, Kurt: Die Bewegungsverhaltnisse in sehr engen Doppelstern
systemen, Schriften der Konigsberger Gelehrten Gesellschaft 10, Heft 4,
1933; or Veroff. Univ. Sternwarte Konigsberg Heft 3.
Many papers dealing with the orbits of particular systems will be found
in Bulletins of the Laws Observatory, the Publications of the Dominion
Astrophysical Observatory, the Contributions from the Princeton University
Observatory, the Astrophysical Journal, the Astronomische Nachrichten, the
Monthly Notices of the Royal Astronomical Society, and other journals and
observatory publications.
The sections on the Algol variables in such works as Die veranderlichen
Sterne, by J. G. Hagen, S. J., and J. Stein, (19131924) and Geschichte und
Literatur der veranderlichen Sterne, by G. Muller and E. Hartwig (1918)
may also be consulted.
R. Prager publishes annually (Kleine Veroff. der Univ. Sternwarte zu
BerlinBabelsberg) a Katalog und Ephemeriden Veranderlicher Sterne, which
includes Algol variables, and T. Banachiewicz, in the Supplements Inter
nationale of the Krakow Observatory Publications, a list of the eclipsing
variables with ephemerides for those of established orbits. In the issues
for 1933, Prager gives elements for 582 eclipsing variables, and Banachiewicz,
light elements and ephemerides for 300, and a list of 722 others, some
which may prove not to be of the Algol type.
CHAPTER VIII
THE KNOWN ORBITS OF THE BINARY STARS
Several hundred orbits of visual binary stars and of stars with
periodic variable radial velocity have been computed by the
methods presented in the preceding chapters. Every computa
tion was undertaken with the immediate object in view of
representing the observed motion and of predicting the future
motion in the particular system on the assumption that the
bodies are moving in obedience to the law of gravitation. It
is satisfactory to find that while the computed orbits exhibit
the utmost diversity in form and in dimensions, we have found
no reason to question the validity of that assumption.
Back of this immediate objective, long since attained, lay the
broader motive of providing additional data for the study of
the greater questions of the origin and evolution of the binary
star systems and of their relation to single star systems. In
the present chapter we shall examine the computed orbit ele
ments, first with respect to correlations that may exist between
them and then more particularly for the information we may
derive from them as to stellar masses and densities.
Not all the orbits which have been computed can be used in
such studies. The observed arcs upon which many of the
orbits of the visual binary stars rest are so short that a great
variety of apparent ellipses may be drawn that will represent
the data within the error of measure. The wellknown binary
a Coronas Borealis offers a striking example: using practically
the same data, Lewis found the period to be 340 years, while
Doberck gave 1,679 years. Even with longer arcs we have, in a
number of cases, two or more radically different orbits. Reject
ing all of these, as well as a few that rest upon assumptions
which seemed plausible but which have not been supported by
later observations, we still find in those retained a wide range
in reliability. The majority are fairly good, some, for all prac
tical purposes, are definitive, a few others only slightly better
than some of those that have been rejected.
203
204 THE BINARY STARS
In Table I in the Appendix, I have listed the 116 pairs retained
(including Cc** of £ Cancri and Aa of £ Ursae Majoris) giving
in general the latest* set of elements when two or more have been
computed for the same pair. The columns of the table give, in
order, the name of the pair, its position for 1900.0, and the
magnitudes and spectral class f (or classes), the orbit elements,
and the authority.
Similar tables of the best available orbits have been published
from time to time, the most recent one being the one by W. H. van
den Bosf). LuplauJanssen and his colleagues at the Urania
Sternwarte, Copenhagen, have also published a catalogue!
containing every orbit published up to 1926. For 21 systems,
10 or more orbits are listed by them, 70 Ophiuchi leading, with
33 different sets of elements.
The orbits of spectroscopic binary stars, based upon measures
of radial velocity, which in every case cover at least one complete
revolution period and as a rule a considerable number of revolu
tions, are, with few exceptions, more accurate than the orbits of
visual binaries. But not every star whose observed radial velocity
varies periodically is a binary system. The observed variation
may be the result of motion of a periodic character in the atmos
phere of a single star. This is apparently true of all those which
also show variation in brightness of the Cepheid type, and also
in general, of those in which the period of variation is but
a fraction of a day, unless they are eclipsing variables like
W Ursae Majoris.
Otto Struvelf has given special attention to stars of this class
and has developed a criterion based upon the relation between K,
the semiamplitude of the velocity variation, and P, the period.
When mean values are used for the quantities involved,
including the masses, this relation  may be written
K = CP~ H (1)
It is simply Kepler's harmonic law in different form, and shows
** = t bnc Cc, the Spectroscopic Binary.
* The tabulation, however, does not. include orbits published after
September, 1934, unless they were available to me in advance of publication.
t From the Henry Draper Catalogue, if the star is listed there.
X B.A.N. 3, 149, 1926.
§ Erganzungshefte zu den A.N. 5, Nr. 5, 1926.
If Ap. Jour. 60, 167, 1924.
 The equation from which Struve derives this relationship is given in the
section on the Masses of the Binary Stars, on p. 218(4).
THE KNOWN ORBITS OF THE BINARY STARS 205
that the semiamplitude should vary inversely as the cube of the
period. Struve finds a very satisfactory agreement between
the observed and the computed values for K for stars of all
periods with the exception of those of two classes : the Cepheid
variables and the stars, not eclipsing binaries, which show
variable radial velocity with periods of but a small fraction of a
day. The mean value of K for 10 stars of this latter class is
very nearly the same as the mean for 15 Cepheid variables
which he investigated, and in neither class do the values vary in
accordance with Kepler's law. Excluding the Cepheid variables
and these short period pseudoCepheids, and also a few stars
of doubtful character, like a Orionis, for which rather uncertain
orbits of long period have been computed, there remain 326 stars
which are undoubtedly spectroscopic binary systems. These
are listed in Table II in the Appendix, which is similar to Table I
in its arrangement. One orbit only is given for each system,
but when the spectra of both components are visible on the plates,
the significant elements for the secondary, if they have been
computed, are also entered. Eclipsing binaries, in this table,
are indicated by an asterisk.
RELATIONS BETWEEN PERIOD AND ECCENTRICITY
Certain striking characteristics of the orbits in the two tables
are recognized on the most casual inspection; for example
the eccentricity of the visual orbits is generally large, that
of spectroscopic orbits generally small; the periods of the
former are long — the shortest known so far (if we exclude
Capella, measured with the interferometer, and pairs like Aa of
£ Ursae Majoris, in which one component is invisible) being
4.56 years (531 = jSIOOOAB);* those of the latter generally short,
ranging, with few exceptions, from a few hours to about 150 days.
See, Doberck, and others have called attention to the high
average eccentricity of the visual binary star orbits and to the
contrast, in this respect, between these orbits and those of the
planets in the solar system. This appears again in Table I.
The average value of e for the 116 systems listed is 0.517 and no
*Kuiper, in his examination of stars of large parallax, has recently
found that the 9.2 star Wolf 390 (= B.D. 8° 4352, 16 h 50™1, 8° 09',
class M3e) is a close pair with components of nearly equal magnitude, and
maximum separation 0"2. His measures indicate that its period of
revolution is less than two years.
206
THE BINARY STARS
less than seven of the individual values exceed 0.90. The
average value for the planets, including Pluto (e = 0.249),
is only 0.08.
On the other hand, the average eccentricity for the 326 orbits*
of spectroscopic binaries in Table II is but 0.174. Recalling the
fact that the periods of the visual binaries are, on the average,
much longer than those of the spectroscopic, it is natural to try
to establish a relation between the two elements. Doberck
long ago presented evidence tending to show that the eccentricity
of the visual binaries increased with the length of the period;
Campbell, Schlesinger, Ludendorff, and others have shown that a
similar relationship exists among the spectroscopic binaries.
If we order the 116 systems in Table I in the Appendix accord
ing to period and eccentricity, we have Table 1.
Table 1. — Periods and Eccentricities of Visual Binary Orbits
~~~^_P, years
050
50100
100150
150 +
Sums
0 10
10 20
7
1
2
1
11
20 30
7
2
9
30 40
7
4
2
1
14
40 50
9
5
4
6
24
50 60
8
7
5
5
25
60 70
2
1
3
6
70 80
4
2
5
2
13
80 90
2
2
3
7
90100
1
1
2
3
7
Sums
47
23
22
24
116
Table 2 gives a similar grouping for 324 of the spectroscopic
binary star orbits listed in Table II, omitting n Orionis and
e Hydrae because they have been counted among the visual
binaries.
The data utilized in these two tables may be summarized as
in Table 3.
These three tables, containing 440 pairs, show practically
the same relations between period and eccentricity as those
brought out by the corresponding tables in the first edition
(drawn up in 1917) when only 187 systems were available for
* Omitting /* Orionis and e Hydrae, which are also visual binaries, the
mean value is 0.173.
THE KNOWN ORBITS OF THE BINARY STARS
207
study. There is an increase of eccentricity with increasing
period, on the average, in the orbits of the spectroscopic binaries
and also in those of the visual pairs, and the average eccentricity
is decidedly larger in the visual than in the spectroscopic binary
orbits. Half the values of e, in Table 1, exceed 0.50, and less
Table 2
"^^P, days
e~\^^
05
510
1020
2050
50100
100500
500 +
Sums
0 10
100
26
19
8
4
10
4
171
10 20
14
10
5
2
10
6
47
20 30
2
9
10
4
3
2
5
35
30 40
1
3
7
3
2
2
5
23
40 50
1
2
1
6
1
3
3
17
50 60
1
2
4
3
1
5
1
17
60 70
1
2
2
2
2
9
70 80
3
1
4
80 90
1
1
90100
Sums
119
53
48
29
13
36
26
324
Table 3. — The Relation between Period and Eccentricity in Binary
Systems
p
n
Av.P
Av.e
0 5 d
119
2^736
0.053
5 10 d
53
7.424
0.155
10 20 d
48
13.551
0.222
20 50 d
29*
29.779
0.358
50100 d
13
69.829
0.234
100500 d
36
229.390
0.279
500 d +
26
, 5? 81
0.286
0 50*
47
27.90
0.444
50100*
23
72.44
0.546
100150 y
22
119.75
0.530
150" +
24
249.02
0.615
♦Eight of these 29 pairs have computed values of e in excess of 0.53; the remaining
21 values range from 0.0 to 0.49, with a mean of 0.249.
than onetenth of all are as low as 0.20; in Table 2, on the other
hand, twothirds of the values of e range from 0.0 to 0.20, and
only 31 of the 324 exceed 0.50.
208
THE BINARY STARS
To bring out more clearly the "scatter" in the values of e
in orbits having approximately the same period, and, simul
taneously, to show that the curve of relationship between P and e
has no break in its continuity from the shortperiod spectroscopic
binaries to the longest period visual binaries for which orbits are
available, we may arrange the data as in Table 4. Here the
spectroscopic binaries are gathered into 11 groups according
to period, and the visual binaries into 9 groups, the successive
columns giving the number of pairs in each group, the mean
period and the mean eccentricity. If the values of e are plotted
against the logarithms of P, and the points are connected we
shall have an irregular broken line, illustrating the scatter in
the values of e; but there is no indication whatever that we are
dealing with more than one relationship curve.
Table 4
N
P
e
30
1^260
0.051
30
2.236
0.038
30
3.248
0.072
30
4.276
0.060
30
6.830
0.125
30
9.528
0.238
30
13.376
0.181
30
23.457
0.356
30
76.090
0.242
30
299.701
0.268
24
2457.36
0.312
13
11?69
0.455
13
24.11
0.489
13
36.89
0.383
13
49.33
0.529
13
77.43
0.519
13
98.44
0.468
13
126 . 91
0.601
13
186.18
0.579
12
308.82
0.627
The results of Russell's statistical investigations of the wider
visual binaries for which no orbits are likely to be available for
many years or even centuries, are of interest in this connection.
THE KNOWN ORBITS OF THE BINARY STARS 209
He finds the average eccentricity for 500 pairs, of average period
roughly estimated at 2,000 years, to be 0.61, and "of nearly 800
more with average period of perhaps 5,000 years" to be 0.76.
We may summarize the data most effectively as follows:
p
n
Av.P
Av.e
to 100 d
262
13 d 5
0.147
100 d +
62
1,023.0(= 2?8)
0.282
Oto lOCF
70
42?5
0.478
100* +
46
187.2
0.574
500
2,000 ±
0.61
800 +
5,000 ±
0.76
But if there is a correlation between the two elements, it is
not a simple one; disturbing factors evidently enter. We note,
for example, the outstanding high average eccentricity for the
29 orbits of spectroscopic binaries with periods ranging from 20 to
50 days.* The value of e exceeds 0.60 in five of these orbits,
as well as in three of shorter period. Again, we note that the
eccentricity of the visual binary OS341,f with a period of only
19.75 years is 0.96, and that in six other visual orbits of period
under 50 years J the value of e exceeds 0.70. Orbits of long period
and small eccentricity are also found both among the visual
and among the spectroscopic binaries. We have, moreover,
to reckon with the fact that the tabulated orbits may to a certain
degree be affected by observational and computational selection
and to that extent fail to be truly representative of binary star
orbits in general. In the case of the spectroscopic binaries this
selection factor may not be very important, although it is easy
to see that systems of long period, and consequently small values
of K, are more easily overlooked than those of short period,
and that of the systems actually discovered, the shortperiod
ones are more likely to be selected for orbit computation.
The situation with respect to the visual binaries may be more
serious, though here it is not so much a matter of discovery as of
observation. If the eccentricity of the orbit is high, the com
* A similar high appeared in the 1917 tabulation which had 13 systems
with periods between 20 and 50 days.
t ADS 11060; cf. T Boo, £186, ADS 9343.
{ In a letter received after this passage was written, Dr. van den Bos gives
reasons for believing that the visual binary OS 536 may have a period of
only 27 years and an eccentricity of 0.98!
210 THE BINARY STARS
panion will spend most of its time at or near the apastron end
of the ellipse and the chances are that the pair will be discovered
when the apparent angular separation is at or near its maximum.
It may then show little orbital motion for a number of years,
particularly if the orbit has a high inclination, and be regarded
by observers as "practically fixed," and therefore be neglected.
Again, a high percentage of the known closer visual binaries
have components that are sensibly equal in magnitude. If
such a pair has an orbit of great eccentricity, it may readily
happen that observers will miss the times of minimum separation
and that the computed orbit will be one of low eccentricity and
of approximately twice the true period. Two systems, 8 Equidei
and £ Scorpii, in which this mistake was actually made, are listed
in Table 1.
Making due allowance, however, for all these and other possible
selective effects, for the actual "spread" in the values of e in
the various period categories, and for the irregularities in the
tabulated progression, it still remains true that there is a general
tendency toward greater values of the eccentricity as the period
increases. This is a fact to be taken into account in theories
of the origin of the binary stars, particularly if we assume that
the spectroscopic and the visual binaries are objects of the same
class.
RELATIONS BETWEEN PERIOD AND SPECTRAL CLASS
Campbell, in his study of the spectroscopic binary stars,
found evidence of a relationship between the period and the
spectral class; taking the spectra in the order B, A, F, G, K,
and M, the period increases as we pass from B toward M.
Before analyzing the present data to see whether they support
this conclusion, it should be said that in combining the various
subclasses, I have taken Class B to include subclasses O to B5;
Class A, subclasses B8 to A3; Class F, subclasses A5 to F4;
Class G, subclasses F5 to GO; Class K, subclasses G5 to
K2; and Class M, subclasses K5 to M6. This agrees with the
Harvard system, except in the inclusion of subclass B8 under
Class A, but differs somewhat from the grouping adopted by
Campbell.*
* Campbell also included a number of systems whose periods were known
to be "long" or "short," though their orbits had not then been computed.
THE KNOWN ORBITS OF THE BINARY STARS
211
Table 5 shows the distribution with respect to period and
spectral class of 439 pairs* entered in the preceding tables.
Table 5. — The Relation between Spectral, Class and Period
"\Spectrum
^\^^
B
A
F
G
K
M
Sum
P ^^\
0 5 d
41
42
16
18
1
1
119
5 10 d
13
23
1
11
5
53
10 20 d
10
18
3
11
6
48
20 60 d
6
8
6
4
6
29
SOrlOO" 1
3
4
5
1
13
ioosoo 4
7
3
3
6
14
3
36
500 d +
4
2
5
14
1
26
0 50"
8
4
26
8
2
47
60100"
4
1
13
5
23
100160"
1
4
3
9
4
1
22
150* +
6
5
7
6
1
2
Totals
85
122
42
114
68
9
440
It will be noted that, while there is a wide range in period in
pairs of every one of the spectral classes, more than twothirds
of the spectroscopic binaries with periods of ten days or less
belong to spectral classes B or A, that pairs with periods in
excess of 100 days are most numerous in Class K, whereas Class M
has, in all, but five representatives. In the visual orbits, on the
other hand, Class G is best represented, especially in pairs with
periods not exceeding 100 years. Class M, again, has few
representatives and but one Class B pair appears.
To investigate the question of progression in period with
advancing spectral class, I have computed the average periods
of pairs of each spectral class, but in doing so have omitted four
spectroscopic binaries of Class B, two of Class A, and one of
Class G, because of their abnormally long periods. The results
are given in Table 6, which records also the average eccentricity
for each group. If the pairs just referred to were retained, the
average period for Class B (84 stars) would become 149^9, for
Class A (100 stars), 36^82, and for Class G (60 stars), 274^20.
The average eccentricities would be but slightly changed.
*The spectral class of the primary star, in visual binaries, is the one
tabulated. The system £ Urs. Maj. Aa is therefore omitted, since the
primary has already been counted.
212
THE BINARY STARS
Table 6. — The Relation between Spectral Class and Average
Period and Eccentricity
Class
Av.P
Av. e
N
B
24?61
0.153
80
A
15.08
0.162
98
F
25.80
0.225
29
G
101.20
0.152
59
K
681 . 12
0.212
46
M
336.07
0.096
5
B
104 y 3
0.314
1
A
120.3
0.565
22
F
133.4
0.566
13
G
84.0
0.518
54
K
97.5
0.478
22
K5
167.45
0.556
2
Ma, b,
43.7
0.296
2
It again appears that the spectroscopic binaries of the "early"
spectral classes have, on the average, short periods, those of the
"late" classes, long periods; but the progression is irregular,
the pairs of Class A having the shortest periods, those of Class K,
the longest. Except for the fact that pairs of Classes G and K
have, on the average, the shortest periods,* the visual binaries
give little evidence of correlation between period and spectral
class.
It has already been pointed out that spectroscopic systems of
short period are more readily discovered than those of long
period, for not only is the amplitude of the velocity curve greater,
in general, in the former, but the variation in the velocity becomes
apparent in a much shorter time. We may expect relatively
more longperiod systems in future discoveries among stars of all
classes of spectra and hence an increase in the average values of
the periods. It should also be noted that the spectra of stars of
the later types, in general, show more lines, and these more
sharply defined, than the spectra of the earlytype stars. The
probable error of measure is therefore less and hence a variable
radial velocity of small amplitude may be unmistakable in,
say, a Class G or K star, whereas one of equal amplitude may
escape* recognition in a star of Class B. This may account, in
part, for the distribution in spectral class shown by the longer
* Unless we count the two M type stars m Herculis BC, and Krueger 60.
THE KNOWN ORBITS OF THE BINARY STARS 213
period spectroscopic binaries in Table 5, but it obviously does
not explain the large number of shortperiod binaries of classes B
and A.
The decrease in the average eccentricity of the visual binaries
with the advance in spectral class from A,F to G,K, is a curious
feature of Table 6. It is apparently a selective effect, for the
percentage decrease is definitely smaller than in the corresponding
table in the first edition which listed 68 pairs. The spectroscopic
binaries show no such progression.
Absolute trigonometric parallaxes are available for 89 of the
visual pairs listed in Table I. I have computed the absolute
magnitudes for the primary components in these systems, and
have divided them into two groups at M = 3.0, which is ordi
narily taken as the point of division between giant and dwarf
stars. Omitting the exceptional white dwarf star, 40 Eridani
BC (M = 10.7), we have the following table:
Table 7. — The Relations between Absolute Magnitude, Spectkal
Class, Period, and Eccentricity
M<L 3.0
M > 3.0
Sp
JV
M
P
e
n
M
P
e
B
1
0.4
104 y 3
0.31
A
19
+ 1.53
111.1
0.58
1
+3.2
139 f3
0.59
F
8
2.28
134.3
0.58
3
3.33
156.3
0.46
G
9
2.26
55.3
0.51
29
4.42
84.6
0.46
K
16
5.71
111.8
0.46
Ma, b
2
11.00
83.2
0.30
The distribution of stars of the different spectral classes in
the two groups occasions no surprise, but it is of interest to note
that fully twofifths of the pairs fall into the group with M ^ 3.0.
In classes F and G, the average P for pairs in this group is smaller
and the average e larger than for pairs in the group M > 3.0;
but these facts, especially in view of the small numbers of pairs
involved, are probably without significance.
THE DISTRIBUTION OF THE LONGITUDES OF PERIASTRON
In 1908 Mr. J. Miller Barr called attention to a singular
distribution of the values of o>, the longitude of periastron, in
those spectroscopic binaries whose orbits are elliptic. In the
214
THE BINARY STARS
30 orbits available to him in which e was greater than 0.0,
26 had values of o> falling between 0° and 180° and only four
between 180° and 360°. He concluded that the effect was due
to "some neglected source of systematic error" in the observed
radial velocities, but both Ludendorff and Schlesinger, examining
the data, were of opinion that it "was nothing more than a
somewhat extraordinary coincidence," for it became less marked
as additional orbits were computed.
In more recent years this question has been discussed by a
number of investigators, some offering theoretical explanations
for the unequal distribution which they regard as real, others
refuting the explanations advanced. The history of these
discussions, with full references, is given by O. Struve and
A. Pogo* in a paper they published in 1929. Their own investiga
tion leads them to the conclusion that the observed distribution
may be real and may arise from conditions in the stellar systems
that would produce a tendency toward a particular orientation
of the periastra with respect to the direction to the center
of the galactic system, but they admit that the evidence is not
conclusive.
I have examined the data in Table II with respect to the dis
tribution of the values of «, with the result given in Table 8.
Table 8
e —
0°90°0
90°180?0
180°270!0
270°360t0
00.10
0.100.20
0.200.50
0.500.88
47
8
28
10
27
13
14
6
22
10
21
7
24
15
13
10
Totals
93
60
60
62
All orbits in Table II, except those definitely noted as circular
and two in which o> was set down as variable, are included. It
appears that the number of values for « in the first quadrant
exceeds by 50 per cent the number in any of the other three
quadrants, but that the excess is most marked in the orbits
for which the value of the eccentricity is not greater than 0.10;
* "Ober die Ursache der ungleichen Verteilung der Periastronlangen bei
spektroskopischen Doppelsternen, A.N. 234, 297, 1929.
THE KNOWN ORBITS OF THE BINARY STARS 215
that is, in those orbits for which the value of oj is least deter
minate. For the orbits with e > .10, the sums, in the four quad
rants, are, respectively, 46, 33, 38, 38, and the excess in the
first quadrant is but 28 per cent. Even this is rather larger
than would be expected in a purely random distribution of values,
but it will be well to wait until a much larger number of accurate
orbits becomes available before accepting it as proof of a real
inequality in the distribution of the periastra.
THE ORIENTATION OF THE ORBIT PLANES OF THE VISUAL
BINARY STARS
A related problem is that of the orientation of the orbit planes
of the visual binary stars. A number of investigations have
been made to ascertain whether these orbit planes exhibit a
random distribution or whether there is a tendency to parallelism
to a particular plane as, for example, the central plane of the
Milky Way.
Practically, the problem is to determine the distribution of the
poles of the orbits, and in its solution we encounter the serious
difficulty that the orbit elements of a binary star do not define
its plane uniquely unless the inclination is 0° or 90°. It is only
when the indetermination in the sign of the inclination has been
removed by spectrographic observations that we can discriminate
between the true and the "spurious" pole. For this reason, the
conclusions reached by the earlier investigators are all open to
question, and it is not surprising that they differ widely. Miss
Everett,* See,f and Doberckf found the distribution to be a
random one: Lewis and Turner § concluded that the evidence
indicated, somewhat doubtfully, a tendency of the poles to
group themselves along the Milky Way; Bohlin^[ noted a division
into two groups, one with a concentration of the poles near the
pole of the ecliptic and the solar apex, the other with a con
centration near the pole of the Galaxy. The more recent
investigations by Kreiken and Shajn** also indicate a con
centration near the pole of the Galaxy.
* Alice Everett, Mon. Not. R.A.S. 56, 462, 1896.
t T. J. J. See, Evolution of the Stellar Systems, 1, 247, 1896.
t W. Doberck, A.N. 147, 251, 1898; A.N. 179, 299, 1908.
§ T. Lewis and H. H. Turner, Mon. Not. R.A.S. 67, 498, 1907.
1f K. Bohlin, A.N. 176, 197, 1907.
j E. A. Kreiken, Mon. Not. R.A.S. 88, 101, 1927.
** G. Shajn, Mon. Not. R.A.S. 86, 643, 1925.
216 THE BINARY STARS
All of these studies rest upon the known orbits of visual binary
stars. Professor J. M. Poor,* on the assumption that parallelism
of the orbit planes would reveal itself as a variation in correlation
between position angle and distance of double stars in different
parts of the sky, based a statistical study on all the data available
in 1913 and concluded that there is a concentration of poles near
the vertex of the preferential motion of the stars.
Quite recently Y. C. Changf and W. S. Finsen $ have investi
gated the question using as data only those orbits for which the
true pole is known. Chang, in 1928, based his study upon
16 pairs, including Capella, and the two systems 42 Comae
Berenices and £ Cawcn,§ whose orbit planes are, respectively,
approximately parallel and perpendicular to the line of sight.
Finsen, in 1933, found 28 pairs available, including five for which
the values of cos i lay between 0.95 and 1.00 or between 0.00
and 0.05. Neither investigator found any striking concentration
of the poles, and both conclude that the distribution is probably a
random one. We may accept that as the best answer to the
question on the basis of existing data.
THE MASSES OF THE BINARY STARS
The only direct method we have of determining the mass of a
celestial body is to measure its effect upon the motion of another
body. It follows that the binary stars are the only ones whose
masses we can determine directly. Since a knowledge of stellar
masses is fundamental in all studies of the dynamics of the
stellar system, the methods by which we calculate the absolute
and the relative masses of the components in the visual and
spectroscopic binaries merit careful attention.
Unfortunately, the orbit elements alone do not afford all
the data necessary for the determination of either mass or
density. The wellknown harmonic law
D 3 :d 3 = P 2 (M + JWi):p 2 (m + mi),
will give the mass of any system in terms of the Sun's mass
when the linear dimensions of the system as well as the orbit
elements are known. But the semimajor axis of the visual
* J. M. Poor, A J. 28, 145, 1914.
t Y. C. Chang, A J. 40, 11, 1929.
% W. S. Finsen, Communicated in manuscript form in August, 1933.
§ Schnauder's orbit.
THE KNOWN ORBITS OF THE BINARY STARS 217
binary stars is known only in terms of seconds of arc, and its
value, so expressed, must be divided by its parallax to reduce
it to linear measure;* and we do not know the true semimajor
axis of the spectroscopic binary orbits at all, but only the
function a sin i. This, however, is expressed in kilometers.
The parallax of a number of visual binaries is known with a
greater or less degree of certainty, and mass values for those
systems may be computed, using the harmonic law in the form
a 3
(m + mi) = ^p 2 (1)
in which r is the parallax of the system, P the period and a
the semimajor axis of its orbit, and the units of mass, length,
and time are, respectively, the Sun's mass, the astronomical
unit, and the year.
While we are unable to derive the mass of any given spec
troscopic binary until we have a knowledge of the value of i,
the inclination, we may nevertheless estimate the average mass
of a number of systems with approximate accuracy, by deter
mining the probable average value of i and hence of sin i. The
formulas required differ for the two cases (1) when both spectra
have been observed, and (2) when only one spectrum is visible.
They may be derived from the wellknown relation
, . 4ir 2 (a + «i) 3 , n s
(w + mi) = *£r pi (2)
in which t denotes, not the parallax, but the circumference of
radius unity, k the Gaussian constant (log 8.23558), a and ai,
the major semiaxis of the orbits of the primary and secondary,
respectively, and P their revolution period expressed in mean
solar days. Since we do not know a but only the function a sin i,
we must multiply both members of (2) by sin 3 i, and since a sin i
is expressed in kilometers, we must divide its value by that of
the astronomical unit expressed in kilometers. The numerical
value of 4ir 2 /k 2 A 3 is approximately f 4/10 20 and we therefore have
, . n • 3 • 4 (a sin i + a x sin i) 3 , ,
(m + mi)sm 3 % = ^ • ^ 2 (3)
* This gives the length in astronomical units. The astronomical unit or
the Earth's mean distance from the Sun is, in round numbers, 149,500,000
km.
f The more precise value 3.99455/10 20 is used in obtaining the logarithm
in Eq. (4).
218 THE BINARY STARS
From Eq. (9) of Chap. VI,
a sin i = [4.13833]#P\/r=T 2
hence
(m + m x ) sin 3 i = [3.01642  10]^ + K X )*P(1  « 2 ) ? * (4)
the numbers in square brackets being logarithms. This equa
tion is independent of the parallax, or distance of the system.
When both spectra have been measured and the correspond
ing velocity curves drawn we obtain at once the relative masses
of the two components, from the relation m:m x = K X :K;
and we also have the equations
m sin 3 i = [3.01642  lORX + K x yK x P(l  e*)*\
m x sin 3 * = [3.01642  10] (K + K X )*KP(1  e*) H j W
from which to compute the masses of the components separately. *
When only one spectrum is visible we must apply a some
what different formula, viz.,
m x 3 ... 4 (a sin i) 3 .„.
sm l = jm — m — C6)
(m + wii) 2 10 20
in which a sin i and m refer to the component whose spectrum
is given. We may write this in a form similar to Eq. (4) thus:
mi 3 sin 3 z = [3 Q1642 _ 1Q]K3 p^ _ eV)H (?)
(m + mi) 2 ' v '
In applying Eqs. (4) and (7) it is necessary to assume a value
for sin 3 i and the question of obtaining such a value has next
to be considered. "It can be shown for an indefinitely great
number of binary systems whose orbital planes are distributed
* It is possible, in the case of certain eclipsing binaries, to determine the
value of the velocity range (/f 2 ) of the fainter star even when the direct
measures of its velocity do not in themselves suffice to define the velocity
curve. It is only necessary to have enough observations to define the slope
of velocity curve of the secondary relatively to that of the primary. From
this relation slope and the orbital elements of the primary, the value of K 2
can be computed, and thus the mass ratio of the two components. Joy, for
example, employed this method in his work on the orbit of U Sagittae
(Ap. Jour. 71, 336, 1930). Here the larger (and fainter) star passes nearly
centrally in front of the smaller primary and the total eclipse of the latter
lasts about 100 minutes. With the 100in. reflector enough spectrograms
were secured in this short time interval to permit the slope of the velocity
curve to be determined with considerable accuracy. The number of stars
in which this method may be employed is, however, small. Contrib. Mt.
Wilson Obs., #401.
[See UOC, #68, Feb. 26, 1926, p. 354.— J.T.K.]
THE KNOWN ORBITS OF THE BINARY STARS 219
at random, that the average inclination would be 57?3, in
accordance with the formula
x X
2 C 2 P
io =  I I * sin idid4> = 1
*"Jo Jo
The average value of sin 3 i, however, would not be sin 3 57?3
( = 0.65) but approximately 0.59 in accordance with the formula
X X
sin 3 i Q =  I I sin 4 idid<f> = Jie* — 0.59"
Campbell, whom we have just quoted, and Schlesinger, who,
from a slightly different formula obtains the same value for
sin 3 to, point out that while this mean value holds for orbits in
general, it would not be permissible to use it for the spectro
scopic binary stars whose orbits have so far been computed.
For, to quote again from Campbell, "there is the practical con
sideration that binary systems whose orbital planes have large
inclinations are more readily discoverable than those whose
inclinations are small . . . Under ordinary circumstances, and
when dealing with a considerable number of orbits, a compromise
value of sin 3 i = 0.65 might in fairness be adopted." For
18 systems which he actually considers he adopts the higher
value 0.75 because six of them are eclipsing binaries, with
inclinations quite certainly between 60° and 90°.
Schlesinger, assuming "that the chance of discovery is pro
portionate to sin i" obtains sin 3 i = 0.68 for a mean value.
We may then adopt, for convenience in computation,
sin 3 i = 0.667 = %,
since comparatively few eclipsing binaries are among the number
under discussion.
Both spectra are visible in 103 of the 321 pairs in Table II
and for these the computers of the orbits have given the values
m sin 3 i, mi sin 3 i, or, for some of the eclipsing binaries, the values
m and w»i. Table 9 lists these pairs with their spectral classes
and mass values. The most massive pair, by far, is BD + 57°28,*
Class B5, and the least massive is probably the eclipsing pair
S Antliae, Class F0, with masses 0.52 O and 0.29 O for the two
components, respectively. Smaller minimum mass values are
* [See p. 251; also cf. Krueger 60, p. 242, a Visual Binary; also (46) Dra,
B4745, Spec. Binary: Lick Obs. Butt. 483, I, #275 (noneclipsing).^J.T.K.]
220
THE BINARY STARS
Table 9. — Masses: Spectroscopic Binaries
2 Spectra
Star
Spec.
m sin 3 i
mi sin 3 t
m
mi
+57?28
B5
113.2
44.9
* Boss 46
B0
17.57
16.37
* TVCas
B9
1.83
1.01
■*• Cas
A5
1.35
1.34
7 And
B3
1.50
1.10
Boss 373
F5
1.16
1.06
k Ari
A0
0.14
0.13
i Tri, br m
GO
1.12
1.12
i Tri fr
F4
0.91
0.86
+59°609
B5
18.88
9.17
Boss 816
B8
2.87
2.76
o Per
Bl
5.42
3.79
A Per
F5p
1.01
0.88
* +33?785
B3
4.86
4.29
Boss 1001
B9
0.56
0.55
+7?676
B5
7.0
3.7
Boss 1213
B9
2.5
2.2
* TT Aur
B5
6.7
5.3
a Aur
GO
1.19
0.94
2 674A
F5
1.40
1.33
Boss 1275
A0
1.71
1.50
* i) Ori
Bl
11.2
10.6
<t> Ori
B2
5.53
4.19
Boss 1457
A0
0.63
0.44
Boss 1464
B2
10.3
3.9
P Aur
AOp
2.38
2.34
3?1413
B5
6.2
4.1
* WW Aur
A0
2.2
1.9
+6?1309
BOp
75.6
63.3
29 CMa
Oe
32.2
24.3
Boss 1906
B8
4.3
2.3
Boss 1945
F5
1.05
0.85
* a Gem C
Mle
0.63
0.57
+34? 1657
F0
1.53
1.32
+20?2153
A0
1.39
1.35
Boss 2484
A0
1.48
1.27
* S Ant
F0
0.52
0.29
o Leo
F5
1.30
1.12
* WUMa
GO
0.67
0.48
Boss 2830
F2, A3
0.28
0.24
uUMa
A0
3.50
0.60
Boss 2987
A2
0.12
0.08
Boss 3138
B3
8.2
4.4
fl'Cru
A5
0.74
0.61
+74?493
G5
0.80
0.70
Boss 3323
A5
4.62
2.37
Boss 3354
A0
2.47
2.08
* RS CVn
F8
1.79
1.66
fUMa
A2p
1.70
1.62
a Vir
B2
9.6
5.8
Boss 3555
F5
2.34
1.92
Boss 3635
F5
1.36
1.29
39 Boo ftr
F5
1.27
1.03
THE KNOWN ORBITS OF THE BINARY STARS
Table 9. — {Continued)
221
Star
Spec.
m sin* t
mi sin' «
m
mi
*UOrB
B8
4.27
1.63
f CrB br.
B8
13.35
13.06
ff Sco
Bl
13.0
8.3
«■ CrB br.
GO
0.94
1.07
+17?3063
A0
2.19
1.35
Boss 4247
F2
1.11
0.99
cHer
A0
1.6
1.0
*UOph
B8
5.31
4.66
* u Her
B3
7.5
2.9
• TXHer
A5
2.04
1.77
Bom 4423
F0
0.88
0.82
+ 14?3329
A3p
1.83
1.62
*Z Her
F5p
1.5
1.3
Boss 4602
F5
0.46
0.41
Boss 4622
F0
1.04
1.01
Boss 4643
A2
1.72
1.18
* RX Her
AO
2.08
1.85
+65?1276
A3
1.97
1.87
+49?2871
F5, A
1.48
1.47
Boss 4788
AO
0.95
0.90
10.4926
B5
7.10
4.43
+ 16.3758
F5
1.26
1.26
*RS Vul
B8
5.26
1.64
USag
B9
6.7
2.0
+37?3413
AO
1.18
0.84
*Z Vul
B3
5.25
2.37
Boss 4947
AO
0.91
0.65
* 9 Aql
B3
5.3
4.4
Boas 6026
F5
1.46
1.44
+35?3970
BO
13.85
12.90
Aql
AO
0.52
0.38
Boss 6173
A2
2.27
2.06
+45?3139
Bl
2.90
2.35
• YCyg
B2
17.4
17.6
Boss 6375
B3
1.79
1.67
+324134
AO
1.87
1.08
+27?4107
FO
0.97
0.77
Boss 5575
A3
1.62
1.54
Boss 5579
AO
0.96
0.95
Boss 5591
AS
1.19
1.17
Boss 5629
B3
20.8
13.6
* RT Lac
G5
1.9
1.0
Boss 6683
F5
0.65
0.61
Boss 5764
B5
0.87
0.71
Boss 5834
B3
6.01
3.87
Boss 5846
GO
1.47
1.38
* +64°1717
B3
11.4
9.8
+58?2546
B3
4.8
2.9
Boss 6142
BO
18.5
12.7
Boss 6148
F5
1.70
1.67
Boss 4745, 46(c)Dn
AO
0.12
0.10
(perhaps the least massive
star known)
The asterisk preceding star names in the first column indicates that the star is an eclipsing
binary.
222
THE BINARY STARS
listed, it is true, but the factor sin 3 i may, in these cases, also
be far below the average value.
It will be noted that two pairs, one in Class B0B2 (+57°28),
the other in Class B3B5 ( + 6° 1309), are extraordinarily massive
and cannot, therefore, be used in deriving mean mass values for
stars of those classes. Omitting them, we have the mean values
given in Table 10.
Table 10. — Mean Mass Values for Spectroscopic Binary Stars
Noneclipsing
Eclipsing
Class
N
1
m sin 3 i
wi sin 3 i
wi/m
N
m sin 3 i
mi sin 3 i
m,\/m
32.2
24.3
0.75
B0B2
10
10.79
8.09
0.75
1
17.4
17.6
1.01
B3B5
16
7.44
4.77
0.64
1
5.25
2.37
0.45
B8A3
27
2.08
1.61
0.77
8
3.75
2.12
0.57
A5F4
11
1.33
1.04
0.78
2
1.28
1.03
0.80
F5G2
18
1.26
1.16
0.92
3
1.24
1.15
0.93
G5
1
0.80
0.70
0.88
1
1.9
1.0
0.53
Mt
1
0.63
0.57
0.93
In compiling this Table, I have taken the mass values for all
stars noted as eclipsing binaries to be the true masses, although
for a number of them the figures in Table 9 are entered as mini
mum values.
Inspection of that table shows that in only two systems
(Y Cygni, B2 and a Coronae Borealis,btr, GO) is mi slightly more
massive than m, and that in only 17 others is there practical
equality between the masses of the two components. The
general rule is that the secondary is definitely the less massive
star. We shall see that this holds true, too, for the visual binaries
for which the mass ratio has been computed.
From Tables 9 and 10 it is clear that binaries of classes O to B5
are decidedly more massive than those of later classes and that
there is a fairly definite progression in the average mass values
as we pass from O to G, though too much stress must not be laid
upon the particular figures in Table 10, since there is a large
range in the individual values for every class.
Conclusions, moreover, that are drawn from systems in which
the spectra of both components are recorded cannot, legitimately,
be extended to all spectroscopic binary systems, for the double
THE KNOWN ORBITS OF THE BINARY STARS
223
line systems are selected, in the sense that it is only in systems
with relatively large values of K that the spectrum of the second
component is visible. The sum (K + Ki) enters by its cube in
Eq. (4) and the mass, therefore, in general increases very rapidly
with K.
The value of the function mj sin 3 i/(m + wii) 2 is frequently
omitted by the computer of orbits for it gives very little definite
information. Equation (7) affords a ready means of computing
the function for any system, but I have not considered it neces
sary to carry out the computation, for a glance at the numbers
recorded in Table II* shows at once that no conclusions could
be based upon any means that might be taken. There is nothing
novel in these conclusions; they simply confirm, on the basis
of more extensive data, conclusions reached by several earlier
investigators. As early as 1911, for example, Ludendorff
found, from the systems then available for study, that those of
Class B were, on the average, about three times as massive as
those of classes A to K.
The last two columns of Table I* give the parallaxes and
masses of 83 visual binary star systems. The parallaxes for the
brighter stars were taken, with but one or two exceptions, from
Schlesinger's Catalogue of Bright Stars; those for the stars too
faint to be listed in that catalogue, from parallax data kindly
sent to me from the Yale University Observatory. It is to be
noted that the latter are not Schlesinger's values but my own
deductions from the data. Taking them all at face value we
have the following summary:
Table 11
Class
JV
(m + mi)
M
B
1
21
11
37
16
2
10.65
4.03
2.59
2.43
2.14
0.62
0.10
B8 to A3
1.04
A5to F3
2.18
F5to G2
3.25
G5 to K2
5.25
Ma, b
10.70
The final column gives the mean absolute magnitudes of the stars.
The means for classes A, F, and G agree well with those for the
corresponding classes of spectroscopic binaries in Table 10. The
* In the Appendix.
224 THE BINARY STARS
numbers for classes B, K, and M are too small to give the means
any weight.
Several of the individual mass values in Table I relate to triple
or quadruple systems. It is known that one component in each
of the systems Ho212 (13 Ceti), A2715 (/* Ononis), k Pegasi,
and 0282, and both components in the systems Castor (AB)
and £ Ursae Majoris (AB) are spectroscopic binaries. Further,
invisible companions have been suspected in the system 70
Ophiuchi and f Cancri (AB) and a fainter companion to the
companion of Sirius has been reported by several observers.
It is not at all improbable that other systems may be found to
have similar additional components when later measures give
us more accurate orbits. In some instances, the mass of the
invisible companion is known to be very small, but even so,
when the data become more extensive and reliable, account must
be taken of all of these extra bodies in any discussion of the
masses of the visual binary stars. At present it will suffice to
call attention to their existence and to the fact that allowance
for them would modify slightly the figures in Table 11. The
uncertainties still attaching to the values of the parallax are,
however, far more important, for many of the parallaxes are
small and changes in their value, within the probable error
of measure, will make large changes in the computed masses
since the parallax enters the formula by its cube. Changes
in the orbit elements P and a will have far less effect for, to a
large degree, they will offset each other, since in general they
vary in the same sense.
It may be remarked that the comparatively small range in
mass in the visual binary systems might have been predicted
from the fact that, in general, longperiod orbits have the larger
values of a, shortperiod orbits, the smaller ones. A large
percentage of the known visual orbits are comparable in size
to the orbits of the major planets in the solar system.
The parallaxes utilized in Table 11 do not depend upon the
fact that the stars to which they relate are binary systems, but
it is to be noted that when, in a visual binary we have not only
the orbit elements but also spectrographic measures of the
relative radial velocities of the two components, we have the
data for an independent determination of the parallax, as See
pointed out many years ago. Hussey used this method to derive
the parallax of S Equulei, and Wright, to derive that of a Centauri,
THE KNOWN ORBITS OF THE BINARY STARS 225
their values agreeing closely with those given in the table. The
computation is readily made by means of the following formulas,
adapted by Wright from the work of LehmannFilh6s:* Let
R = the astronomical unit, expressed in kilometers.
a = the semimajor axis of the binary, expressed in kilo
meters, and a", the same element expressed in seconds
of arc.
n = the mean angular motion of the star, in the visual orbit,
in circular measure per second of time.
AV = the observed difference in the radial velocity of the two
components.
Then
2tt
n ~ 86400 X 365.26 X P
Ayvr^? \ (8)
n sin i[e cos w + cos (v + «)J
a
The micrometric measures connecting the two components of
a visual double star afford data for the computation of the
relative orbit only, and give us no direct information about the
position of the center of gravity of the system or the mass ratio
of the two components. This information can be obtained only
from measures connecting one, or both, of the components with
independent stars. Such measures, covering a sufficient time
interval, afford the data for the computation of the absolute orbit
of the component concerned, and thus, by comparison, of the
mass ratio.
The classic illustration is the star Sirius. Bessel, in his dis
cussion of the proper motion of the bright star, based upon the
meridiancircle observations, noted as early as 1834 that it was
not moving uniformly along a straight line but was describing
a wavy line across the sky. He inferred the existence of an
unseen companion, the two components revolving about their
common center of gravity in a period of fifty years, f The
faint companion discovered by Alvan G. Clark in 1862 fully
confirmed this prediction, and the combination of the micrometer,
* I have made slight changes in Wright's notation as given in L.O.B. 1,
4, 1904.
f For a more detailed statement, see p. 237.
226 THE BINARY STARS
meridiancircle, and parallax measures permits the computation
of the mass of each component and also of the linear dimensions
of the orbits.
When a series of micrometric or photographic measures con
necting the components of a binary with an independent star is
available, the relative masses can be determined in a very simple
manner.*
Let AB be the binary system, C an independent star, and
let p, 6 and p', 0', respectively, be the distance and position
angle of C referred to A and of B referred to A. Then the
apparent rectangular coordinates of C and B referred to axes
drawn from A as origin in position angles O and (90° + O )
will be
x = p cos (0 — do)
x' = p' cos (0' — do)
y = p sin (0 — Bo)
y' = p' sin (6' do)
Now if we let K equal the mass ratio B/(A + B), the coordinates
of the center of gravity of AB will be Kx r , Ky', and since the
motion of C with respect to this point must be uniform, we have
x = a + b{t  to) + Kx') y = a' + V(t  t ) + Ky', (9)
to being any convenient epoch.
Each set of simultaneous observations of AB and AC fur
nishes an equation of condition in x and one in y for the deter
mination of the five constants a, 6, a', &', K. No knowledge
of the period or other elements of the binary system is involved,
the accuracy of the determination of K depending entirely
upon the amount of departure from uniformity of motion of B
relatively to A. In Lick Observatory Bulletin 208 I published
a list of systems specially suited to the application of this method
and urged the desirability of measuring them systematically.
The late Lewis Boss deduced the mass ratios for a number
of systems and published his results as an appendix to his
Preliminary General Catalogue of Stars for 1900.0. Taking his
values and a few obtained by other investigators and applying
them to the masses for the systems, as given in Table I,f we have
the following data on the masses of the components in visual
binary systems :
* Russell, Ap. Jour. 32, 363, 1910.
f In the Appendix.
THE KNOWN ORBITS OF THE BINARY STARS
Table 12
227
Star
n Cass
40Erid. BC
Sinus
Procyon
f Cancri AB
6 Hydrae AB
f Urs. Maj. Aa, Bb
7 Virginis
a Centauri
£ Bootis
THerc
Melb4, AB
n Here BC
70 0ph
Krueger 60
86Pegasi
* [K Pegasi AB (A is a Sp. Bin.) .
Spec.
F8
A
A0
F5
GO
F8
GO
F0
GO
G5
GO
K2
Mb
KO
Ma
GO
F5
wii/wi
0.76
0.45
0.39
0.36
1.00
0.86
1.00
1.00
0.85
0.87
0.45
0.75
0.80
0.78
0.67
1.86
0.60
0.72
0.44
2.40
1.17
1.24
2.02
0.83
1.19
0.95
0.73
1.22
0.67
0.46
0.80
0.24
0.35
3.30
(A+a)
wii
0.55
0.21
0.96
0.43
1.23
1.74
0.83
1.19
0.81
0.63
0.52
0.51
0.37
0.72
0.16
0.65
1.90
^J.T.K.]
The evidence available at present leads to the conclusion that
the fainter component in the system 85 Pegasi is the more massive
one, but Boss considered the uncertainties so great that he
finally gave the two components equal masses. The results for
the other systems indicate that the fainter star is the less massive
one, a conclusion which is in harmony with that derived from the
spectroscopic binary stars.
DYNAMICAL + PARALLAXES OF THE VISUAL BINARY STARS
Equation (1), page 217, written in the form
a
where p is the parallax, may obviously be used' to compute the
parallaxes of systems with known orbits for which a value of the
masst can be determined or assumed from independent data;
but the number of such systems is small and is not increasing
very rapidly. It has, however, been known for many years
that two or more sets of elements for a given system may have
but little resemblance to each other as a whole and yet give
* [IAck Obs. Bull. 483, Tab. I, #342 R, p. 22, Luyten, 1934.—J.T.K.]
t The term "hypothetical" was used in earlier years to describe these
parallaxes.
J The Sun's mass is, as usual, taken as the unit in this discussion.
228 THE BINARY STARS
approximately the same value for the ratio a*/P 2 , which enters
our equation. The reason is, as Jackson says, that the various
sets of elements "give nearly the same arc for that portion of the
orbit which has been observed and that this arc is sufficient to
determine the gravitational attraction between the two stars."
In fact, in order to define the relation between the parallax
and the mass of a system, all that is needed is an observed arc
long enough for the computation of a satisfactory value of the
double areal velocity p 2 (ddldt).
Comstock* was the first, apparently, to apply this principle,
using it to derive the masses of systems in slow orbital motion
for which parallaxes were available. Russell, f Hertzsprung,J
and Jackson and Furner§ later developed formulas for deter
mining dynamical parallaxes on the basis of assumed average
values for the mass m x + m 2 . Hertzsprung, in 1911, adopting
the value (mi + w 2 ) = 1, derived "minimum hypothetical"
parallaxes and concluded that, statistically, the ratio of the
true to the minimum hypothetical parallaxes, does not vary
greatly and can be expressed by
log^ = +0.27 ± 0.14
Ph.min.
Jackson and Furner, the first to publish an extensive list of
dynamical parallaxes, used (ra x + m 2 ) = 2 for systems with
orbits, and l/\/mi f m 2 = 0.855 for the much larger number
with observed arcs too short for the computation of even pre
liminary elements.
In 1923, Russell, Adams, and Joy^[ published their comparison
of 327 dynamical parallaxes, computed by Russell on the assump
tion that the mass of each system equals that of the Sun, with
the spectroscopic parallaxes derived at Mount Wilson. They
grouped the stars according to spectral class, separating giants
*Publ. Washburn Obs. 12, 31, 1908.
t Determinations of Stellar Parallax, A.J. 26, 147, 1910; Science (N.S.)
34, 523, 1911.
t Vber Doppelsterne mil eben merklicher Bahnbewegung, A.N. 190, 113,
1911.
§ Jackson and Furner: The Hypothetical Parallaxes of 556 Visual Double
Stars, with a Determination of the Velocity and Direction of the Solar
Motion, Mon. Not. R.A.S. 81, 2, 1920.
If A Comparison of Spectroscopic and Dynamical Parallaxes, Publ. A.S.P.
36, 189, 1923.
THE KNOWN ORBITS OF THE BINARY STARS 229
from dwarfs from F6 to M, taking absolute magnitude = 3.0
as the dividing point. They also computed the absolute magni
tude for each pair, both from the dynamical and from the
spectroscopic parallaxes. The relationship between absolute
magnitude and mass was striking; stars of all spectral classes
and giants as well as dwarfs, "fell into line." "It is evident,"
they wrote, "that statistically considered, the mass of a binary
system is a function of its absolute magnitude." The theoretical
explanation of the massluminosity relationship, has been given
by Eddington. In his more recent work on dynamical parallaxes,
Russell* has developed formulas which include a factor, n,
depending upon the massluminosity relationship and has drawn
up tables from which its value may be obtained.
For pairs with orbits, his formula is
p = ndx = naP~ H
and for physical pairs with only a short observed arc, he writes
h = nh lf hi = l^£ = 0.418^^
where 8 is the observed distance, w the observed relative motion
in seconds of arc a year, and I a factor (derived from a statistical
discussion not given in detail) which is designed to make the
mean values of d x and h x equal. The values of d x and hi (except
for the factor of proportionality) are derived directly from the
data of observation. To find n, three tables, A, B, and C, are
provided. Table A gives the correction to be applied to the
observed visual magnitude of a pair to reduce it to the "stand
ardized bolometric magnitude to." This correction, of course,
depends upon, and varies with, the spectral class (that of the
primary star is taken in general) or assumed temperature. The
bolometric absolute magnitude, M 1} then follows from
M i = to + 5 + 5 log d x (for pairs with orbits)
« to + 5 + 5 log hi (for other physical pairs)
With M x as argument, take out n from Table B, and with Mi
and AMi the difference of magnitude between the two com
ponents, as arguments, take A from Table C. Then
n = rioA. (1)
* On the Determination of Dynamical Parallaxes, A. J. 38, 89, 1928.
230
THE BINARY STARS
Table A
Sp.
Temp.
Corr.
Sp.
Temp.
Corr.
Sp.
Temp.
Corr.
BO
23000°
0?9
gGO
5600°
0?0
dGO
6000°
+0?1
B5
16000
0 .1
gG5
4700
0.4
dG5
5600
.0
AO
11000
+0.2
gKO
4200
0.7
dKO
5100
0 .2
A5
8600
+0 .3
gK5
3400
1 .5
dK5
4400
0.6
FO
7400
+0 .3
gM2
3100
2 .0
dMO
3400
1 .5
F5
6500
+0 .2
gM7
2700
2 .6
Table B
Mi
n
Mi
Wo
Mi
Too
6
0.065
0.417
6
0.880
5
0.097
1
0.496
7
0.961
4
0.132
2
0.575
8
1.046
3
0.189
3
0.652
9
1.138
2
0.251
4
0.728
10
1.238
1
0.337
5
0.S03
11
1.345
Table C
AM
Mi = 4
2
+2 and fainter
1.00
1.00
1.00
1.00
1
1.02
1.01
1.01
1.01
2
1.05
1.04
1.03
1.03
3
1.08
1.07
1.06
1.05
4
1.11
1.09
1.08
1.07
5
1.13
1.12
1.11
1.10
6
1.14
1.13
1.12
1.12
7
1.15
1.14
1.14
1.14
8
1.16
1.15
1.15
1.15
9
1.16
1.15
1.15
1.16
10
1.16
1.16
1.16
1.18
12
1.17
1.17
1.18
1.19
In 1929, Russell and Miss Charlotte E. Moore published the
dynamical parallaxes of 1,777 double stars,* derived by the
method just described, and in 1933, Miss Moore computed the
dynamical parallaxes of 323 of the pairs of my own discovery,
from observational data which I provided, f
* A.J., 39, 165, 1929.
t Lick Obs. BvU. 16, 96, 1933, #451.
THE KNOWN ORBITS OF THE BINARY STARS 231
Mr. R. 0. Redman,* in a paper published just before Russell's,
also gave the results of an investigation planned to improve
dynamical parallaxes by making the masses conform to the
massluminosity relation instead of adopting a mean standard
mass. His discussion differs from Russell's in its details and
results in more laborious processes for the actual computations
of parallax. Finsenf has recently made an instructive compari
son of the formulas developed by Comstock, Jackson, Hertz
sprung, and Russell, and has added a formula for use in the case
of binaries whose periods are known, though only a small actually
observed arc is available. Several pairs of this kind are given
in double star catalogues.
Dynamical parallaxes are, statistically, of a high order of
accuracy, particularly when the massluminosity relation has
been taken into account, but it does not follow that the dynamical
parallax of an individual star can be taken as the measure of
its distance. Comparison of the dynamical parallaxes computed
by Russell and Miss Moore with the parallax values entered in
Table I,J for example, shows that the mean of the differences is
only +0"00016, but individual differences range from +0"009
to — 0"008, with a few, for the nearer stars, that are much
larger.
DENSITY OF THE BINARY STARS
Although every shortperiod spectroscopic binary star would
be an eclipsing binary to an observer in the plane of its orbit,
as Stebbins and others have remarked, the methods outlined in
Chap. VII to determine the density of an eclipsing binary
from the orbit elements cannot be applied to spectroscopic
binaries in general, and still less to the visual binaries. But
from the relations that have been established theoretically
between the absolute magnitude, diameter, and temperature of
a star, confirmed as they have been in many critical cases by
interferometer measures and by the investigations on eclipsing
binaries, it is possible to determine the diameters and the masses
of stars, and hence their densities.
* A Statistical Study of the Effect of the MassLuminosity Relations on
the Hypothetical Parallaxes of Binary Stars, Mon. Not. R.A.S. 88, 33, 1927.
fThe Determination of Dynamical Parallaxes of Double Stars, Mon.
Not. R.A.S. 92, 47, 1931.
% Appendix.
232 THE BINARY STARS
The fact that a star is a component of a binary system has no
bearing either upon the principles involved in these methods,
or upon their application and for this reason they are not treated
here. It may be remarked, however, that the evidence now
favors the conclusion that the densities of stars of the main
sequence, from Class B to Class G, whether they are single
stars or components in a spectroscopic or visual binary system,
are very similar to those found for the eclipsing binaries. There
are, of course, a few exceptions among components of binary
systems, as, for example, the very dense companion to Sirius
which will be discussed in detail in Chap. IX.
MULTIPLE STARS
In 1781, Herschel noted that the brighter star of the 5" pair,
f Cancri, discovered by Tobias Mayer in 1756, was itself a
double star with an angular distance of only 1" between its
nearly equal components. In the years that have followed, a
large number of such triple systems, and not a few that are
quadruple, or multiple, have become known. During the Lick
Observatory double star survey, for example, I catalogued at
least 150 such systems previously unknown, and Prof. Hussey's
work yielded a proportionate number. The triple was formed,
in more than half of these cases, by the discovery of a close
companion to one of the components of a wider pair previously
catalogued by other observers, and in some cases there is no
question but that the closer pair had been overlooked at the
earlier date because it was below the resolving power of the
telescope.
The spectrograph has also revealed many triple and mul
tiple systems; sometimes, as in 13 Ceti or k Pegasi, by show
ing that one component of a visual binary is itself a binary
too close to be seen as such with the telescope; again, as in
Algol, by showing that the shortperiod spectroscopic binary
revolves in a larger orbit with a third invisible star.
In 1918 I estimated that 4 or 5 per cent of the visual binaries
are triple or quadruple systems if components known only
spectroscopically or through the perturbations they produce are
included as well as visible additional companions. The compila
tion of the New General Catalogue of Double Stars offered oppor
tunities to check that estimate. I did not make a complete
count, but was confirmed in my opinion that this estimate is
THE KNOWN ORBITS OF THE BINARY STARS 233
very little, if at all, too liberal. It seems to be a general rule
that the distance between the components of the close pair
in such systems whether visual or spectroscopic is small in
comparison with that which separates the pair from the third
star. However, there are exceptions to the rule. Thus we
have in Hu 66, BC = 0'.'34, A and BC (= 02 351) = 0'/65; in
A 1079, AB = 0'23, AB and C = 0'/48; in A 2286, AB = 0"34,
AB and C = 0''94; in A 1813, AB = 0'.'20 AB and C = 0'.'70;
and in Hu 91, BC = 0"15, AB (= OS 476) = 0"54. Some
allowance must, of course, be made for the effect of perspec
tive; the orbit plane of the closer pair may not coincide with
the plane in which the third star revolves. But it is unlikely that
this will modify the relative apparent angular distances greatly.
The system of Castor affords an extreme example of the
contrasting distances between the close and wide pairs in a
quadruple star; each component of the visual pair is a spectro
scopic binary, the revolution periods being respectively three
and nine days while the period of the orbit described by these
two pairs is certainly greater than 300 years! The motion of
the third star with respect to the closer pair in a triple visual
system has in no instance been observed over an arc long enough
to permit the computation of a reliable orbit.
The various conclusions drawn in the present chapter as to
masses and densities of the binary stars and as to the relations
between the orbit elements, rest upon comparatively small
numbers of pairs, but some of them, none the less, may be
accepted as definitely established. Others may require modifica
tion, when additional data become available.
References
From the many papers published in recent years on subjects discussed
in the present chapter, I have selected the following, in addition to those
quoted in the footnotes, as representative.
Ludendorff, H.: Weitere Untersuchungen liber die Massen der spektro
skopischen Doppelsterne, A.N. 211, 105, 1920.
Hektzsprung, E. : On the Relation between Mass and Absolute Brightness
of Components of Double Stars, B.A.N. 2, 15, 1923.
Jackson, J. : The Masses of Visual Binary Stars of Different Spectral Types,
Mon. Not. R.A.S. 83, 444, 1923.
Shajn, G. : The Movement of the Line of Nodes in Spectroscopic Binaries
and Variables and its Consequence, Ap. Jour. 67, 129, 1923.
Eddington, A. S.: On the Relation between the Masses and Luminosities
of the Stars, Mon. Not. R.A.S. 84, 308, 1924.
234 THE BINARY STARS
Brill, A.: Strahlungsenergetische Parallaxen von 123 visuellen Doppel
sternen, Veroff. Univ. Sternwarte BerlinBabelsberg, 7, I, 1927.
Lundmakk, K. : Statistical Concerning the Binary Stars, Ark. f. Mat. Astr.
och. Fysik, 20A, 12, 1927.
Pitman, J. H.: The Masses and Absolute Magnitudes of Binary Stars, A. J.
39, 57, 1929.
Shajn, G.: On the MassRatio in Binary Stars and the Hypothesis of a
Secular Decrease of Mass, A.N. 237, 57, 1929.
Krbiken, E. A.: Many papers: e.g., Man. Not. R.A.S. 89, 589, 1929; B.A.N.
4, 239, 1928; 6, 71, 1929; A.N. 238, 373, 1930.
CHAPTER IX
SOME BINARY SYSTEMS OF SPECIAL INTEREST
Having studied the orbit elements of the binary stars in
their more general relations, it will be of interest next to con
sider the various systems in themselves, the extent, and the
limitations, of our knowledge of their motions and physical
conditions. Selection is here an obvious necessity, and in
making my choice I have been influenced in part by the his
torical associations connected with certain systems, in part
by the peculiarities of the orbit. Some of the systems are
among those for which our knowledge is relatively full and
exact; others present anomalies still more or less baffling to
the investigator.
a CENTAXJRI
Our nearest known stellar neighbor, a Centauri, is a system
of more than ordinary interest. One of the first half dozen
double stars to be discovered — the very first among the stars
of the southern heavens — it also divides with 61 Cygni the
honor of being the first whose approximate distance, or par
allax, became known. It consists of two very bright stars,
0.3 and 1.7 magnitude, respectively, which revolve in a strongly
elliptic orbit so highly inclined to the plane of projection that at
times they are separated by fully 22", at others by less than 2".
Accurate micrometer measures of relative position begin
only with Sir John Herschel, in 1834, but meridian circle obser
vations date back to Lacaille's time, 1752. Since these early
dates the system has been observed regularly with meridian
circle, micrometer, and heliometer, and the position of its com
ponents has been measured on photographic plates. The
material is therefore ample for a very good determination of
the orbit elements and of the proper motion of each component
and excellent use has been made of it by Roberts, See, Doberck,
Lohse, and, in 1926, by Finsen, whose elements are quoted in
Table I. Finsen's period is 1.25 years longer than that found by
235
236 THE BINARY STARS
Lohse, but his other elements differ little from the earlier set and
we may regard his results as definitive. The parallax is known
with equal precision; the value resulting from the excellent
heliometer measures by Gill and Elkin having been confirmed
by later discussions of meridian observations by Roberts and
others and by Wright's results from measures of the relative
radial velocities of the components, to which reference has been
made on an earlier page. Schlesinger* gives the value 0"760
and assigns the proper motions — 3"604 in right ascension and
H0"739 in declination to the system of the two stars. The
spectrograph has also given us the radial velocity! of the center
of mass of the system, —22.2 km/sec.
Taking Finsen's elements and the parallax value just quoted,
we find that the semimajor axis of the system is 23.2 A.U.,
but since the eccentricity of the orbit is 0.52, the distance between
the components at periastron is but 11.2 A.U. only a little
greater than Saturn's mean distance from the Sun, whereas at
apastron it is 35.3 A.U., a value about midway between those of
Neptune and Pluto from the Sun.
In 1904, when Wright measured the radial velocities of the two
components of a Centauri, he found that the brighter star was
approaching the Earth with a velocity of 19.10 km/sec, and the
fainter one, with a velocity of 24.27 km/sec. That is, relatively
to the center of mass of the system, the primary was receding from
the Earth, the companion approaching it. The companion's
position angle at that date was approximately 207°, and the
nodal point of the orbit is 25?4; hence, on the system of notation
we have adopted (Chap. IV), the sign of the inclination is
negative.!
The mass of the system corresponding to the adopted values
of the parallax and orbit elements is 1.96 times that of the
Sun, and all investigators of the proper motions of the two
components agree that the brighter star is very slightly the more
massive of the two. Besides being practically equal to the
Sun in mass, it belongs to the same spectral class (GO) and has
nearly the same absolute magnitude, 4.73 as compared with
4.85. It is therefore almost a replica of the Sun. The fainter
component belongs to Class K5, and its absolute magnitude is
* Catalogue of Bright Stars, p. 110, 1930.
t Moore, Publ. Lick Obs. 18, 115, 1932.
% It was incorrectly stated to be positive, in the first edition.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 237
6.10. Combining the values for proper motion and radial
velocity, we find a space velocity of 25.2 km/sec, but little greater
than that of the velocity of our Sun.
A new chapter in the story of this system was opened when
Innes, in 1915, discovered that "Proxima Centauri," a faint
star of 10.5 apparent magnitude (15.0 absolute) had approxi
mately the same proper motion as the bright star, although a
little more than 2° distant from it. It proved also to have very
nearly the same parallax, and is no doubt physically connected
with the two bright stars though too distant to affect their orbital
motions. "Proxima" is a little nearer to the Sun (4.16 light
years) than a Centauri (4.30 light years) and is therefore our
nearest known stellar neighbor.
snuus
Several references have been made to Sirius on the earlier
pages of this volume, but it will not be amiss to give a more
connected account of the star here. It was in 1834 that Bessel
noticed that the proper motion of Sirius, the brightest star in
the sky, was variable. Six years later he noted a similar
phenomenon in the proper motion of Procyon, and by 1844 he
had worked out the nature of the variation sufficiently to
become convinced that it was due in each instance to the attrac
tion of an invisible companion. His famous letter to Humboldt
on the subject has often been quoted: "I adhere," he wrote,
"to the conviction that Procyon and Sirius are genuine binary
systems, each consisting of a visible and an invisible star. We
have no reason to suppose that luminosity is a necessary property
of cosmical bodies. The visibility of countless stars is no argu
ment against the invisibility of countless others."
Peters examined the existing meridian circle observations in
1851 and concluded that they supported Bessel's hypothesis;
ten years later, T. H. Safford repreated the investigation and
"assigned to the companion a position angle of 838 for the
epoch 1862.1." The most complete discussion, however, was
that of Auwers, who "placed the question beyond doubt by
determining the orbits and relative masses of the bright star
and the invisible companion; but before the results were pub
lished, Mr. Alvan G. Clark discovered the companion, in 1862,
near its predicted place." Bond's measures for the epoch
238 THE BINARY STARS
1862.19, in fact, placed the companion 10'.'07 from the primary
in position angle 84 6.
Since that time it has described more than an entire revo
lution and the orbit elements, now known with high precision,
agree as well as could reasonably be expected with Auwers's,
computed before the companion's discovery. Thus, Volet's
orbit, computed in 1931, which differs very little from my own,
published in 1918, has the revolution period 49.94, whereas
Auwers gave 49.42 years.
The eccentricity of the true orbit is greater than that for the
orbit of a Centauri, but the inclination of the orbit plane is
considerably less and the apparent ellipse is therefore a more
open one, the maximum apparent separation of the components
being about 11"2 and the minimum a little less than 2". The
bright star is so exceedingly brilliant, however, that it is impossi
ble to see the faint companion with any telescope when it is near
its minimum distance. Thus, periastron passage occurred
early in 1894, but the last preceding measure was Burnham's
in the spring of 1892 when the angular separation was 4'' 19,
and the little star was not again seen until October, 1896, when
my first measure gave an apparent distance of 3" 81. The
photometric magnitude of Sirius (and also its photographic
magnitude, since it is a star of Class AO) is —1.6. Estimates
of the apparent magnitude of the companion are subject to great
uncertainty because they are all affected by the presence of the
intensely brilliant primary. Visual observers, in recent years,
have adopted the value 8.5, and Wendell's value from measures
with a doubleimage Rochon prism photometer* is 8.44. This
value has recently been confirmed by Kuiperf from measures
made with the Leiden 10in. refractor, fitted with objective
gratings. On the other hand, Vyssotsky,J using the 26in.
refractor of the McCormick Observatory and (a) coarse objective
gratings, and (6) a rotating sector, derives the value 7.1 for the
photovisual magnitude.
The question of the precise magnitude of this star is an impor
tant one, because of its relation to the question of the star's
* Harvard Annals 64, No. VI, 1909.
t B.A.N. 6, 197, 1932. More recently, Kuiper using the 12in. refractor
at the Lick Observatory and suitable gratings, found the value 8.42, in full
agreement with his Leiden result (Publ. A.S.P. 46, 99, 1934).
t Publ A.S.P. 42, 155, 1930; Ap. Jour. 78, 1, 1933.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 239
density. The mass of the system corresponding to the parallax
and orbit elements in Table I is 3.36 times that of the Sun and
this, with Boss's value for the mass ratio,* 0.39, gives 0.94O
for the mass of the companion. The presence of the bright
primary makes accurate determination of the spectral class
difficult, but Adams finds that it is a little earlier than F0, and
it is now usually called A7. It is the best known representative
of the peculiar group called "white dwarfs." Taking the
apparent magnitude as 8.4, and assuming the companion to
have the surface temperature of a normal star of its spectral
class, the density is found to be nearly 50,000 times that of water.
The density corresponding to Vyssotsky's apparent magnitude
determination, 7.1, is only about onesixth as great.
It was pointed out by several writers a number of years ago
that a star of great density should produce an "Einstein shift"
in its spectrum far greater than the one predicted in the spectrum
of the Sun. In a single star this prediction cannot be tested
observationally for there is no way to distinguish between an
Einstein shift and the shift arising from the star's radial velocity.
In a double star system, like that of Sirius, however, for which
we know the parallax, the orbit elements and the mass ratio,
the radial velocity of the faint component for any date can be
computed from the observed radial velocity of the primary.
Allowance can then be made for it and any residual Einstein
shift detected. The brilliance of Sirius makes the observations
of the spectrum of its companion extremely difficult, as I have
already noted, but Adams f with the 100in. reflector at Mount
Wilson secured spectrograms, the measures of which, after
allowance for the radial velocity and the blend effect from the
scattered light of the bright star, gave a displacement correspond
ing to +19 km/sec, and Moore, J using the Mills spectrograph
attached to the 36in. Lick refractor, later found precisely the
same displacement. The theoretical value, according to Edding
ton, is +20 km/sec.
These observations evidently support the greater value for
the density of the companion and hence the value 8.4 for the
apparent magnitude, but further investigation is desirable.
* Preliminary General Catalogue, p. 266, 1910.
t Proc. Nat. Acad. Set. 11, 382, 1925.
t Publ. A.S.P. 40, 229, 1928.
240 THE BINARY STARS
Accepting the value 8.4 for the apparent magnitude, it follows
that the primary is 10,000 times as bright as its companion,
whereas it has only two and a half times its mass and is less
than 1/50,000 part as dense. And yet the two stars, presumably,
had a common origin. No satisfactory theory has been advanced
to account for such a system. If we adopt Vyssotsky's value,
7.1, the contrasts are not so great, but the difficulties in the way
of an explanation are not lessened.
There is another question relating to this system that must be
considered: Is a third body present? Fox, on one night in 1920,
suspected one less than a second of arc from B, the Clark com
panion, and van den Bos, Finsen, and other observers at the
Union Observatory felt certain that they saw such a companion,
about a magnitude fainter than B, on several nights in 1926,
1928, and 1929. On other nights, however, when they rated the
definition as equally good, they did not see it, and Innes, in
publishing the observations,* added that "in view of the negative
evidence, it would be wrong to assert that the companion C
exists without any doubt."
On the negative side, it is also to be said that neither Burnham
nor Barnard, both noted for remarkable keenness of vision, saw
such a companion at any time in the course of their many meas
ures of the system with the 36in. Lick and 40in. Yerkes refrac
tors. My own eyes are not so keen as theirs were, but I must
add that I have never seen one, though I have measured the
system annually since 1896, often under the most favorable
observing conditions. Although negative evidence is always
less convincing than positive testimony, I think it fair to return
the Scotch verdict "not proven" on the question of the direct
observation of this companion.
There is, however, another method of investigating the
problem, and that is to ascertain whether the measures of the
system AB give evidence of perturbations that might be explained
by a third body. Zagarf and Volet, t to name only the two
most recent writers on the subject, have made thorough investiga
tions using different methods and both agree not only that such
* The Observatory 52, 22, 1928.
f F. Zagar, II terzo corpo nel sistema Sirio, R. Oss. Astr. Padova Nr. 23,
1932.
% Ch. Volet, Recherche des perturbations dans le systeme de Sirius,
Bull. Astron. 8, 51, 1932.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 241
perturbations exist, though of small amplitude, but also give
approximately the same hypothetical orbit for the disturbing
body, on the assumption that it is a companion to B. Unfor
tunately, the computed position angles of C, the third body,
differ from those observed in 1920, 1928, and 1929 by amounts
ranging from 70° to 170°. The 1926 observation is the only
one in reasonable agreement with the theory. It is, however,
impossible to say definitely, as Volet points out, that the hypo
thetical close companion attends component B; it may attend
the brilliant primary A, in which event it would be quite impossi
ble to observe.
KRUEGER 60
The system known as Krueger 60, though the closer pair was
really discovered by Burnham in his careful examination of all the
double stars noted by Krueger in the course of his meridian
circle observations, offers a strong contrast to the two we have
been considering, not only in its appearance but in many of
its physical characteristics, but like them it is remarkable for
its large proper motion and its large parallax. Sirius and
a Centauri are two of the brightest stars in the sky and are
also of great absolute brilliance; Krueger 60 is only of the ninth
magnitude, despite its large parallax, and is among the feeblest
of known stars in its actual radiating power.
The companion has now made very nearly one complete revolu
tion since its discovery in 1890, and since 1900, when Doolittle
called attention to the rapid orbital motion, has been very well
observed, Barnard, in particular, making a remarkable series of
measures — often on 18 to 21 nights a year — extending from 1900
to 1921. The orbit elements, therefore, while not so accurate
as those of Sirius, are very well known. The parallax of the
system is even better known, and the proper motion is also
accurately determined.
It happens that several independent stars are in the same
field of view with Krueger 60, one, of 9.6 magnitude, being less
than 30" distant in 1890, and but little over 1' now. Since the
apparent orbit of the binary is a fairly open one, the minimum
angular distance between the components exceeding 1"4, the
conditions are specially favorable for the determination of the
mass* ratio of the two components (see Chap. VIII). Our
knowledge of the physical conditions in the system is therefore
242 THE BINARY STARS
far more complete than is the case for the average system with
good orbit elements.
The parallax, 0"257, and my orbit elements computed in 1925
(P = 44.27 years, a = 2 / '46)give the mass 0.45 O ; using Huffer's
later elements (P = 44.52 years, a = 2''362) and the same
parallax, the computed mass is 0.40 O. These are probably the
limiting values. Alden,* in 1925, found the mass ratio to
be 0.835. With Huffer's elements, this gives the mass of A
as 0.220, that of B, 0.18O. The latter, which is probably
correct to within one or two units of the second decimal place,
is the smallest mass so far established for any star.
Estimates of the magnitudes, as is usual in double star systems,
vary considerably. The combined magnitude is given as 9.0
in the BD, and 9.1 in the Astronomische Gesellschaft Catalogue,
but on the International (photovisual) Scale, Kuiper finds it
to be 9.64, and the difference of magnitude of the two components,
1.56. This would give 9.9 and 11.4 as the apparent magnitudes
of A and B, respectively. The corresponding absolute magni
tudes are 11.9 and 13.4. According to Adams, the spectrum of
A is Mb, and it is probable that the spectrum of B is of even
later type. Both stars therefore belong to the class of red
dwarfs, and the fainter one is one of the least luminous stars
known. We may agree with Russell that they are nearing "the
very end of their evolutionary history."
It is worth noting that Krueger 60 is one of the eight known
visual binary systems within five parsecs or about 16 lightyears
distance. It is because it is so near us, relatively speaking, that
we know anything about it. The mean parallax for stars
of apparent magnitude 9.0 is, according to Seares, 0"0039.
Remove Krueger 60 to the corresponding distance (more than
800 light years) and it would be beyond the resolving power of
any existing telescope, and the combined image of the two
components would be of about the eighteenth magnitude.
61 CYGNI
There are a number of wide double stars whose components
have large proper motions of nearly the same amount. If the
directions of the motion are different, the measures soon enable us
to decide whether the pair is a physical, or, like 2634 (ADS 3864),
merely an optical one. But when the directions differ but little
* Pop. Astron. 33, 164, 1925.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 243
and the apparent relative motion is small or nearly rectilinear,
it is not always easy to classify the pair on the basis of the
micrometer measures alone. Consider, for example, S1321
(ADS 7251). The two components,* both 8.1 magnitude, have
an angular separation of a little less than 20" and the relative
motion in 92 years has been but 23° and 1"4. When the posi
tions of B with respect to A are plotted, they lie along a straight
line within the error of measure. But the large proper motions
differ so little that we may adopt the value 1"683 in 247*6 for
both components, and the parallax, +0"165, is the same for
each. Unquestionably they constitute a physical system.
This is also true of 34 Groombridge (ADS 246), though the
two components, separated nearly 40", differ in brightness by
more than two magnitudes and their relative motion in 50 years
is but 3?5 and less than 1"0, for they have the same large parallax,
+0"282, and proper motion, 2V89 in 82°5.
Other examples might be given (e.g., E2398 = ADS 11632t),
but the best known pair of this class is undoubtedly 61 Cygni,
(22758 = ADS 14636) which is famous also as the first star for
which an approximately correct value of parallax was determined.
This bright pair (magnitudes 5.57, and 6.28) has been known
since the time of Bradley who, as quoted by Sir John Herschel,
gave the position for the date 1753.8 as 54°36' nf 19''628. Her
schel listed the measures made to the end of the year 1822 and
argued from them and from the large proper motion assigned to
the pair by Piazzi and Bessel that the two stars constitute a
binary system J as otherwise in the course of "nearly 70 years,
during which they have been observed, one of them would
doubtless have left the other behind, without supposing a coin
cidence too extraordinary to have resulted from accident."
Nor did he fail to point out that the data make "61 Cygni a
fit object for the investigation of parallax."
This early argument notwithstanding, the question of the
character of the pair long remained an open one, for it developed
that the proper motions of the two components are not quite
identical, Auwers giving the values 5" 191 in 5152 and 5" 121
in 53°68 for A and B, respectively, and the numerous observa
tions since 1830 for many years gave little evidence of departure
* [+53° 1320/21, HD 79210/11, in U Ma.J.T.K.]
t [+59° 1915, in Draco.J.T.K.]
% Observations of 380 Double and Triple Stars in the Years 1821, 1822 and
1823, by J. F. W. Herschel and James South, p. 367, London, 1825.
244 THE BINARY STARS
from relative rectilinear motion. C. F. W. Peters,* it is true,
computed a set of orbit elements in 1885, finding a = 29'' 48,
P = 782.6 years, and e = 0.17, but the observed arc was entirely
too short for accurate orbit computation and as late as 1905
Burnham clearly intimated that he thought the pair an optical
one. In that same year, however, Osten Bergstrand, from
a thorough examination of all available data, including a special
series of photographs taken by himself in the years 1899 to 1903,
demonstrated the physical relationship of the two stars by show
ing not only that they have the same parallax, but also that
the path of B relatively to A is concave, f This was confirmed
by Schlesinger and Alter, in 19104
More recently provisional sets of elements have been computed
by P. Baize § and by Alan Fletcher, f The former, using graphical
methods, finds a nearly circular orbit (e = 0.013) with a period
of 756 years and a semimajor axis equal to 32 / '8. The latter,
adopting a mass value of 1.126 from the massluminosity relation
ship (Baize's elements give 2.28) and using special analytical
methods, finds an orbit of marked eccentricity (e = 0.404)
with a period of 696.63 years and a semimajor axis equal to
24"525. This shows how wide the range of possible solutions
still is. It is probable that the components are nearly equal
in mass, but whether they are each equal to the Sun or have
only half its mass is still in doubt. After another half century
it will be possible to reach more definite conclusions.
The system is, in any event, one of vast dimensions, its orbit
having a semimajor axis at least twice as great as that of Pluto,
but it must be emphasized that orbits of this size are by no means
exceptional. There are certainly far more visual binaries with
orbital dimensions of this order than there are of systems like
S Equulei, which by way of contrast, we shall next consider.
8 EQUULEI
Until it had, recently, to yield its place to 531, to which
Dawson in his preliminary orbit computation assigns a period
* Bestimmung der Bahn des Doppelsterns 61 Cygni, A.N. 113, 321, 1886.
t Untersuchungen tiber das Doppelsternsystem 61 Cygni, Nova Acta R.
Soc. Set. Upsaliensis, Ser. IV, Vol. 1, n. 3, 1905.
X See Chapter IV, page 121.
§ Le Systeme double de 61 Cygni, Bull. Soc. Astron. de France 41, 20, 1927.
f The Binary System 61 Cygni, Mon. Not. R.A.S. 92, 121, 1931.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 245
4.66 years*, 5 Equvlei was the visual binary of shortest known
period. This, of course, excludes Capella, which has been
measured with the interferometer, and the pair £ Ursae Majoris
Aa, one of whose components is invisible, but is known not
only from the variable radial velocity of A but also from the
perturbations it produces in the relative orbit of the pair AB.
For many years, the period of 8 Equuhi was supposed to be
11.4 years, and the orbit nearly circular, for the two components
are nearly equal in brightness, and the pair is below the resolving
power of existing telescopes at times of minimum angular
separation. Hussey, in 1900, however, showed conclusively
that the period is only half as long, 5.7 years, and the orbit a
fairly elongated ellipse (e = 0.39). Another system resembling
6 Equvlei in that the two components had been mistaken for
each other after passing the point of minimum separation, is
£ Scorpii AB. The accepted orbit, prior to 1905, had a period
of about 105 years and an eccentricity of about 0.13, but my
measures with the 36in. refractor in 1904 and 1905 proved this
to be incorrect. I found that the true period is only 44.7 years
and the orbit really a very eccentric one (e = 0.75).
Returning to S Equulei, we note that the orbit elements are
well determined, the period in particular, since the pair has
made 14 revolutions since its discovery by Otto Struve in 1852.
The semimajor axis is 0"27 and this, since the paraUaxf is 0"066,
corresponds to about 4 A.U. The orbit is therefore decidedly
smaller than that of Jupiter and resembles the larger and more
eccentric minor planet orbits. The mass of the system is 2.11
times that of the Sun and the two components probably are
nearly equal in mass. Each is therefore approximately equal
to the Sun in mass, but since the absolute magnitudes of the two
are, respectively, 4.4 and 4.5 and the spectral class is F5, they
are brighter than the Sun and shine with a whiter light.
TWO MULTIPLE SYSTEMS
Two of the best known double stars in the northern heavens
are Castor (a Geminorum), discovered by Bradley and Pound
* See footnote, p. 205, for Kuiper's discovery of a visual binary with period
of less than 2 years.
t Schlesinger's Bright Stars. Maxwell (Publ. Lick Obs. 16, 311) found
0 048 from the orbit and the relative radial velocity of the components.
This gives a = 5.6 A.U., which is a little larger than that of Jupiter.
246 THE BINARY STARS
in 1719, and £ Ursae Majoris, discovered by Sir William Herschel
on May 2, 1780. Both pairs are among those used by Herschel
in the papers, presented to the Royal Society in 1803 and 1804,
in which he gave observational as well as theoretical reasons
for his belief that many double stars are physical systems, with
components in orbital motion. £ Ursae Majoris was also the
first, and Castor perhaps the second pair for which an orbit was
computed. Savary, in 1830, used the former system to illustrate
his method of orbit computation,* and Sir John Herschel, in
1833, used both pairs, among others, to illustrate his very
different method. The orbital motion in £ Ursae Majoris is
far more rapid than that in Castor, and for that reason these
early orbits computed for it were more successful than Herschel's
orbit of the latter pair.
Struve began to measure £ Ursae Majoris in 1826 and from that
time on the visual observations have been numerous and well
distributed. The two components differ but one magnitude in
brightness, and the angular distance between them, even at
minimum, is but little less than one second of arc. The pair is
therefore also well suited for photographic measurement, and,
in fact, the photographic measures are more accurate than the
visual ones, as appears from the series made at Potsdam by
Hertzsprung and Munch in the years 1914 to 1923, and from the
one by Przybyllok, at Konigsberg in the following years.
Ultimately, an orbit of a system of this character based entirely
upon photographic measures will probably be more accurate
than one resting upon visual observations, but it is not so certain
that modern photographic measures used in combination with
earlier visual ones, particularly if the latter were made with small
telescopes, will give a better orbit than visual measures alone,
unless very great care is exercised in making the combination.
Van den Bos did not overlook this point when, in 1928, he
computed the orbit of £ Ursae Majoris, which has now described
nearly two revolutions since Struve's first measures, and his
elements, which differ but little from those of Norlund (1905),
may be regarded as practically definitive. He gives P = 59.863
years, e = 0.4128, and a = 2"5355.
If this were a simple binary system there would be no occasion,
other than its association with the early methods of orbit com
putation, to single it out for special notice. It is, however, not
* See Chap. IV.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 247
a binary, but a quadruple system. Wright* in 1900, noted that
the radial velocity of the brighter star is variable, and Norlund,
in his investigation of the orbit of the visual pair in 1905, dis
covered a perturbation with an amplitude of 0"05 and a period
of about 1.8 years. This he attributed to the presence of a
third body, but he was apparently unaware of Wright's discovery.
Wright, in 1908, showed that the radialvelocity observations
confirmed Norlund's period, for which van den Bos gives the more
precise value, 1.8321 years. In a sense, as Hertzsprung has
remarked, this may be called the shortest period established for
a binary star on the basis of micrometrical observations, f This
shortperiod orbit proves to be elliptic, (e = 0.50) and to be
highly inclined to the plane of projection, the inclination,
whose sign is fixed by the radial velocity measures, being — 84°5.
The unseen companion, a, is therefore in retrograde motion,
about A, just as B is. Apparently, the general rule is that the
wide and close pairs in a triple or quadruple system revolve in
the same direction, but this is not without exception. It is
not followed in four of 21 systems which van den Bos examined.
In 1918, the Lick observers announced that the fainter com
ponent, B, of the visual system is also a spectroscopic binary of
short period. BermanJ has recently made a thorough investiga
tion of the orbit of this pair and finds it to be circular, with a
revolution period of only 3.9805 days. The orbital inclination
is probably too small to permit even a partial eclipse, and van
den Bos finds that, notwithstanding its high inclination, the
pair Aa is also not an eclipsing binary.
Van den Bos investigated the question of the distribution of
masses in this system; with the advantage of the knowledge of
the orbit of Bb, Berman has repeated the investigation. Taking
the value 0"126, the mean of the 10 modern results, for the
parallax, and van den Bos's elements of the visual pair, he
finds the total mass to be 2.27 times that of the Sun. Adopting
0.77 as the best value the present data will give for the mass
ratios, the mass of Aa is 1.280, of Bb, 0.990. Taking van den
Bos's value for the mass ratio a/A he finds the mass of A to be
0.93 O , that of a, 0.35 O . The mass ratio b/B may have, accord
ing to Berman, any value between x /z and }£.
* Ap. Jour. 12, 254, 1900.
t Prior to 1935.
i Lick Obs. BuU. 16, 109, 1931, #432.
248 THE BINARY STARS
The combination of two such unlike shortperiod binaries
as the components of a 60yearperiod system presents a problem
of unusual interest to the student of the origin of the binary
stars, as Berman has not failed to remark.
Turning next to Castor, we find an even more remarkable
multiple system, for not only are the bright stars, A and B, of
the historic double, the visible members of shortperiod spec
troscopic binaries, but a third star, of the ninth magnitude,
73" distant in 165°, which has the same parallax and proper
motion as the bright pair, is also a shortperiod binary. The
system Bb was discovered by Belopolsky, at Pulkova, in 1896,
the system Aa, by Curtis, at Lick, in 1904. Curtis's investiga
tions* showed that B and its faint companion revolve in prac
tically circular orbits in a period of 2.928285 days, whereas A
and its invisible companion travel in elliptic orbits (e = 0.5033)
and have a period of 9.2218826 days. Curtis, moreover, on the
basis of the observed data and what seemed to be reasonable
assumptions, concluded that the fainter system Bb was about
six times the more massive one. Luytenf has recently given
reasons for adopting the more moderate mass ratio of two to
one as the maximum possible, but even that is an anomalous
result, for there are few exceptions (and those not above suspi
cion) to the rule that in binary systems, both visual and spectro
scopic, the brighter component is the more massive one. Luyten,
himself, regards the true mass ratio as still unknown, and a
careful review of all of the published data leads me to the same
conclusion.
In 1920, Adams and Joyt announced the fact that the ninth
magnitude companion to Castor, which we may call Castor C,
is a shortperiod binary, and in 1926 Joy and Sanford§ computed
its orbit from the measures of radial velocity. Before their
paper was sent to press, van Gent's paperlf appeared, with an
orbit of the pair as an eclipsing variable based upon photometric
observations. The combined data show that the components
of Cc revolve in circular orbits with an inclination of 86°4 in a
period of 0.814266 day. It also appears  that the brighter
* Lick Obs. Bull. 4, 55, 1906.
t Publ. A.S.P. 45, 86, 1933; Publ. Minn. Obs. 2 (1), 313, 1934.
t Publ. A.S.P. 32, 158, 1920.
§ Ap. Jour. 64, 250, 1926.
1[ B.A.N. 3, 121, 1926.
 See Table 1, Chap. VII, p. 200.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 249
component, C, has a mass 0.63 O, the fainter one, c, a mass
0.57 O, that the densities of the two are, respectively, 1.40 and
1.80 O, their radii, 0.76 O and 0.68 O and the distance between
their centers 2.7 X 10 6 km. In other words, our knowledge of
this latest known of the four orbital systems in this great multiple
group, three spectrographic and one visual, is by far the most
accurate and complete.
Our knowledge of the first known of the group, the visual
binary pair, on the other hand, is the most incomplete. The
majestic scale of the relative orbit, with a semimajor axis nearly
twice as great as the mean distance of Pluto from the Sun, is,
of course, responsible. Many hundreds of measures have been
made since 1826, when Struve began his series, but in the interval
to 1931 the companion had described an arc of but 53°. The
earlier measures, back to 1719, give only the direction of B
from A, with occasional rough estimates of distance. It is
now evident that the point of maximum separation in the third
quadrant was passed some 25 or 30 years ago, and the range of
possible orbit solutions is therefore much smaller than it was in
1900, but orbits with periods ranging from 340 to 477 years
still represent the latest measures about equally well. As so
frequently happens, however, the mass of the quadruple system
AaBb is better known than the orbit elements. It is less than
was thought some 20 years ago, and is probably about five
times the mass of the Sun.
It may be noted that while there is no question of the physical
relationship of the distant system Cc to the bright quadruple
system, there is no hope at all of securing evidence of its orbital
motion about the center of gravity of AaBb for many centuries,
for, on the most favorable assumptions, the revolution period
must be of the order of 25,000 years.
CAPELLA
In 1899, Campbell, at the Lick Observatory, and Newall,
at the University Observatory, Cambridge, England, inde
pendently discovered the fact that Capella is a spectroscopic
binary star, with the spectra of both components visible. Camp
bell's announcement was published in the Astrophy steal Journal
for October, 1899, and Newall's, in the November, 1899, number
of the Monthly Notices, R.A.S. In March 1900, Newall* pub
* Mon. Not. R.A.S. 60, 418, 1900.
250 THE BINARY STARS
lished a paper on the binary in which he showed that the period
is approximately 104 days, that the two components are nearly
equal in mass and not very unequal in brightness and argued
that the inclination of the orbit exceeded 27°. Reese,* in 1901,
published a more detailed investigation, based upon spectrograms
secured at the Lick Observatory in the period September 1, 1896
to September 27, 1900. His value for the period is 104.022 days,
for the eccentricity, 0.0164, and for K, the semiamplitude of
the velocity curve of the primary star, 25.76 km. He gave
1.26 as the approximate value for the ratio K^/K 2 = M l /M 2 .
Interesting as these facts are, they would not, taken alone,
call for special comment. But Anderson, f in 1920, successfully
applied the interference method of measurement developed by
Michelson to observations of Capella, using a new type of
interferometer which he had himself devised, and these measures
were continued later in that same year and early in 1921 by
Merrill who utilized the entire series for the computation of an
orbit of the system, % adopting, however, Reese's period, which
Sanford§ had shown to be very nearly correct. He found the
orbit a little more nearly circular than Reese gave it, his value
for e being 0.0086.
Important results arose from the combination of the inter
ferometer with the spectrographic orbit data. This permitted
the derivation of the values of a and *', separately, fixed the
algebraic sign for i, and, most important of all, gave an extremely
accurate value of the parallax of the system, and hence also
values of the masses of the two components and of the separation,
in linear measure, of their centers (ai + a 2 ). These values are:
a = 0"05360, i = 41°08, (a x + a 2 ) = 126,630,000 km
7T = 0"0632, mi = 4.2 O, m 2 = 3.3 O
The visual magnitude of Capella is 0.21 and since the two
components differ in brightness by 0™5, their magnitudes 1[ are,
respectively, 0.74 and 1.24, which, with the given parallax,
correspond to the absolute magnitudes —0.26 and +0.24.
Both components are therefore giant stars.
* Lick Obs. Bull. 1, 32, 1901.
t Ap. Jour. 61, 257, 1920.
t Ap. Jour. 56, 40, 1922.
§ Publ. A.S.P. 34, 178, 1922.
If According to the accurate formula, not the approximate values tabu
lated on p. 54.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 251
The maximum angular separation of the components measured
with the interferometer is 0"0550, a fact that fully accounts for
the failures to separate the two components with the most power
ful telescopes. Interferometer measures, however, are made
visually, and, in a sense, we may therefore refer to Capella as
the visual binary of shortest known period.
Merrill's value of the parallax of Capella is probably as accurate
as any stellar parallax value we have. This fact, and the
completeness and accuracy of our knowledge of the other data
relating to the system, led Eddington to adopt the brighter
component, which belongs to spectral class GO, as the foundation
for the numerical application of his theory of the Internal Con
stitution oj the Stars, including the massluminosity relation.
Assuming the effective temperature to be 5200°, Eddington finds
the absolute bolometric magnitude of this component to be
CT36, which is 5.26 magnitudes brighter than the Sun. From
these values he derives the total radiation of the star (127 X O),
its radius (13.74 X O), and its mean density 0.00227 gm per
cubic centimeter and then proceeds to further conclusions as to
the star's internal structure and temperature, which do not
come within the scope of this volume.
THE MASSIVE BINARY HD 698 ( = BD. + 67° 28)
As long ago as 1904, Hartmann,* in examining the spectro
grams of 5 Ononis (Class B0), secured at Potsdam in the years
1900 to 1903, discovered that the calcium line K differed from
the other lines in the spectrum (mostly due to hydrogen and
helium) in two important particulars. It was narrow and
sharp, whereas the other lines are more or less diffuse, and it
gave a nearly constant velocity, with a mean value about equal
to the velocity of the center of mass of the system, instead of
showing as the other lines did, variable velocity with a range of
about 200 km/sec. The calcium line, apparently, did not
originate in the star.
In explanation, Hartmann assumed the existence of a calcium
cloud stationary in space (at least so far as radial velocity is
concerned) lying between us and the star. This was plausible,
since the Orion region is known to be one filled with nebulous
matter and the mean velocity, +18.7 km/sec, from the calcium
line was nearly the same as that of the Sun's velocity of recession
from the Orion region.
* Ap. Jour. 19, 268, 1904.
252 THE BINARY STARS
Questions, however, soon arose, for it appeared — particularly
from Slipher's researches* — that the calcium lines in other stars
of the Orion type, that is, of classes O to B2, widely distributed
around the sky in low galactic latitudes, exhibited similar
anomalous phenomena, and Miss Heger (Mrs. C. D. Shane),
in 1919 and 1921, showed that, in some of these stars at least,
the sodium D lines behaved in like manner.
The theory now accepted to account for the observed facts
is the one advanced by Eddingtonf in 1926. He postulated the
existence of a uniformly distributed interstellar cloud of extremely
low density, and showed that it was competent to produce the
observed absorption in the spectra of all stars distant enough to
give sufficient depth to the cloud. More recent evidence,
based upon direct observation, has been given by O. Struve,J
Gerasimovic,§ and particularly, Plaskett and Pearce,1f that
practically establishes the fact of the existence of Eddington's
hypothetical cloud. For it appears that the intensities of the
interstellar lines increase with the distance of the stars, that the
cloud, like the stars, rotates about the galactic center, and that
the mean distance of the cloud is about half the average distance
of the stars.
One difficulty remained. While it was not to be expected that
the interstellar lines of calcium and sodium would show in the
spectra of stars of classes F to M, since they would be masked
by the broad, diffuse stellar lines, it was hard to explain why
they did not appear in the spectra of at least some of the binaries
of Class A in which the range in variable velocity is large.
Pearce's investigation of the system HD 698, of spectral class
B9sek and visual magnitude 7.08 removes this difficulty.
Merrill and Humason,** in the course of their study of stars
with bright hydrogen lines, found the radial velocity of this
star to be variable. Their announcement was made in 1925,
and Pearce;}! has now computed the orbit of the pair, on the
basis of spectrograms secured at the Dominion Astrophysical
*LoweU Obs. Butt. 2, 1, 1909.
t Bakerian Lecture: Diffuse Matter in Interstellar Space, Proc. Roy. Soc.
A, 111, 424, 1926.
t Ap. Jour. 66, 163, 1927; 67, 353, 1928.
§ Ap. Jour. 69, 7, 1929.
% Mon. Not. R.A.S. 90, 243, 1930.
 It may be well to explain that the letter s signifies sharp lines, the letter
e, emission lines, and the letter k, the presence of interstellar calcium lines.
** Ap. Jour. 61, 389, 1925.
* Mon. Not. R.A.S. 92, 877, 1932.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 253
Observatory. He finds the spectrum of the primary to be that
of a star of high intrinsic luminosity for Class B9, and that of the
secondary (apparently a normal star of Class B5) to be relatively
faint and measurable only at times of maximum separation
of the lines. The interstellar calcium absorption line, K, is also
present and was measured on 18 of the 40 plates.
Before commenting upon the K line let us note the orbit
elements derived from the measures of the lines for the primary
star. The period is 55.904 days, the eccentricity 0.033, neither
value being in any way exceptional. The semiamplitudes of the
velocity curves, too, while large (K = 85.5 km, K x = 215.5 km),
are not extraordinary, a statement that may also be made of the
mass ratio they give, m 2 /mi = 0.40. But the minimum mass
values are remarkable, msm 3 i being 113.2 O, and m 2 sm 3 i,
44.9 O. These are the largest mass values so far found in any
binary system, and it is to be remembered that the inclination
is less than 90°, since the pair is not an eclipsing variable, and
that the true masses therefore are even greater than those
given.
Turning now to the calcium line K, we note that Pearce found
a blend of the interstellar line with the stellar K line of the
primary through about 41 days of the 56day period, the blended
velocities varying from +38 km/sec to 52 km/sec. But
for a few days when the stars were passing through the nodes,
the three lines, from the primary star, the secondary star, and
the interstellar source, were clearly separated, and the interstellar
lines from 12 observations at these favorable times gave a con
stant velocity of 13.9 km/sec. The velocity of the center
of mass of the binary system is 24.5 km/sec. These results
for a Class A star afford strong support for Eddington's theory
of the origin of the interstellar lines in spectroscopic binary
star spectra.
ALGOL *
Persei, or Algol, as it is more commonly called, has been
known as a variable star since 1670, when Montanari not only
noted the fact of variation in light but actually observed it at
* [Kopal, Zdenek; A Study of the Algol System, Ap. Jour. 96 (3), 399420,
^Elem^nts of the Third Orbit: P: 1.873 yrs.; e: 0.26; i (provisional): 72°;
Mass: 1.0 + 0.30.
Semimajor axis of relative orbit (a t + a,): 36 x 10' km; apparent separa
tion at maximum elongation: O'.'IO. — J.T.K.]
254 THE BINARY STARS
minimum brightness* on November 8, but the general character
of its light variation was first established by Goodericke, in 1783.
He found that the successive minima occurred at intervals of
about 2 20 49 m , the descent to, and recovery from, minimum
covering about eight or nine hours, and that for the rest of the
period the light remained sensibly constant. He explained the
phenomena by assuming that a dark companion, revolving
about a common center with Algol, produced a partial eclipse of
the bright star once in each revolution, but this hypothesis,
though essentially correct, as we now know, was not generally
accepted and, in fact, was almost forgotten, until Pickeringf
revived it in 1880, and Vogel,t in 1889, found the star to be a
spectroscopic binary, with a period equal to that of its light
variation.
The star has been the subject of so many memoirs that it would
be impossible to give even a full reference list of them here, to
say nothing of an adequate account of the work they represent
or of the theories advanced in them. We must limit ourselves
to a general description of the more significant advances made
in our knowledge of the system.
Argelander§ was the first to demonstrate the existence of
fluctuations in the period between successive light minima,
and Chandler's H more extensive studies, utilizing all available
observations from Goodericke's time to 1888 (and in his later
work to 1897), not only confirmed this conclusion but led him to
explain them as arising from a longperiod inequality which he
ascribed to the influence of a third body in the system.
His formula for the period failed, however, to represent later
epochs of minima, and that has also been the fate of more
recently derived formulas, including Hellerich's (1919). There
still remain small periodic and irregular variations not fully
accounted for by theory.
Belopolsky's discovery, in 1906, of a variation in the radial
velocity of the center of mass of the eclipsing system in a period
of the order of 1.8 years, was the next step in advance. The
work of Curtiss in 1908, and, more conclusively, that of Schles
inger in 1912, established the existence of the third body called
* See Porro's note, A.N. 127, 41, 1891.
t Proc. Amer. Acad. 16, 1, 371, 1880.
tPubl. Potsdam Obs. 7, 111, 1889.
§ BonnBeob. 7, 343, 1869.
If A.J. 7, 165ff, 1888; 22, 39, 1901.
SOME BINARY SYSTEMS OF SPECIAL INTEREST 255
for by this variation. Curtiss gave 1.899 years for the revolution
period, Schlesinger, 1.874. The latter also found the semi
amplitude of the velocity curve to be 9.14 km, and the orbit to
be nearly circular. The revolution of the eclipsing pair about the
center of mass in this longperiod orbit, produces an oscillation in
the times of minimum light, which, as Schlesinger points out
may amount to a displacement of five minutes in either direction.
Meanwhile, Stebbins had perfected his selenium photometer,
so that in the years 1909 and 1910, he was able to measure the
brightness of stars to the third magnitude with far greater
accuracy than had been possible by visual or photographic methods.
Applying it to the study of Algol, he discovered the existence of
a secondary minimum in its light curve, with a depth of but
0.06 magnitude and showed, moreover, that the light varied
continuously between minima. Ten years later he repeated the
investigations with the far more sensitive photoelectriccell
photometer. The new light curve was, naturally, more accurate
than the earlier one, but confirmed the secondary minimum and
the continuous variation in the light. It also showed an effect
resulting from the ellipsoidal shape of the components. The orbit
of the eclipsing system was shown to be practically circular.
The rotation of a star upon its axis has the effect of broadening
the lines of its spectrum unless the axis is directed toward us
since one limb is receding, the other approaching us relatively
to the motion of the star's center. This effect, which has received
special attention from O. Struve in his studies of the emission
lines in Class B stars, is particularly pronounced in the case of
eclipsing binary stars, since the receding limb of the component
entering eclipse is the visible one, whereas the approaching
limb is the visible one on emerging from eclipse. Rossiter
studied this effect in j8 Lyrae, and McLaughlin, in Algol, in
1923 and 1924. In both cases it was found to be well marked,
the residuals before minimum light in Algol, with which we are
here concerned, all being positive and those after minimum
negative, as theory requires. The observed range of the effect
was 35 km. Using Stebbins's values for the relative dimensions
of the system and for the inclination of the orbit to compute the
range, McLaughlin found it necessary to assume the brighter
star to be five times as massive as the fainter one to secure
agreement with the observed value. On this assumption, with
the observed range of the rotation effect, it becomes possible to
256 THE BINARY STARS
compute the absolute dimensions in the system, the mass and
density of each component, and also the parallax.
Combining McLaughlin's spectrographs results with Steb
bins's photometric ones we have the following data for the
eclipsing system:
Period (Hellerich's value) 2^867301
Duration of principal minimum 9* 66
Inclination of orbit (cos i = 0.142) 81? 8
Distance between centers (ab + a/) 10,522,000 km
Radius of bright body 3.12©
Radius of faint body 3 . 68 O
Mass of bright body 4 . 72 O
Mass of faint body . 95 O
Density of bright body 0. 16 O
Density of faint body . 02 Q
The corresponding dynamical parallax of Algol is +0'031.
It is specially to be noted, however, that nothing is yet known
of the spectrum of the third star or of any effect it may have
upon the photometric measures of the eclipsing pair.*
McLaughlin, in his closing paragraphs, refers to the difficulty
of obtaining a reliable trigonometric parallax for Algol, because
of its revolution in an orbit about twothirds as large as the
Earth's orbit in a period of less than two years. This is a point
to which Wright called attention as long ago as 1904, in his
note on the parallax of a Centauri and which he elaborated more
fully in 1921 in his notef "On Spectroscopic Binaries and the
Determination of Parallax." It is, obviously, one to be specially
noted by parallax observers.
References
It is impossible^ and, happily, unnecessary to list here the many papers
that have been consulted in the preparation of this chapter. Footnote
references have been given to a number of the more important ones, and the
extensive catalogues of double stars, of spectroscopic binary stars, and of
variable stars will help any reader who wishes to look up more fully any of
the systems that have been described.
* It should be noted that M. A. Danjon in his photometric study of
Algol (Annates, Strasbourg Obs. 2, 148, 1928) reaches conclusions differing
in many particulars from those described above. These conclusions,
however, await confirmation.
jPubl. A.S.P.Z3,47, 1921.
X The references to the literature on Algol alone would fill four quarto
pages of fine print! It is hardly necessary to add that not all of these
papers were read in preparing my note.
CHAPTER X
STATISTICAL DATA RELATING TO THE VISUAL BINARY
STARS IN THE NORTHERN SKY
The visual binary stars for which orbits have been computed
or in which the observed arc suffices for the computation of
dynamical parallaxes may be utilized, as has been shown in
Chap. VIII, in the study of the relations between the orbit
elements and for the investigation of stellar mass. It is obvious,
however, that even if they were not selected stars, they would
not afford an adequate basis for a study of the number of the
visual binaries, of their distribution (apparent or real), or of
the absolute magnitudes of their components. The data required
for a thoroughly satisfactory investigation of these and similar
problems would include a complete enumeration of all binary
systems to a definite limiting stellar magnitude, and measures
(a) of the angular distances of all pairs for a given epoch, (6)
of the photometric magnitudes and spectral classes of both
components, and (c) of the parallaxes of the systems.
The first step in an approximation to this ideal material is
to make a survey of all stars in the sky to a definite magnitude
with sufficiently powerful telescopes used under good observing
conditions. All previously known pairs must be noted and all
new pairs falling within predetermined limits of magnitude and
angular separation identified and measured. The Lick Observa
tory survey and the surveys of the southern sky which are still
in progress at Johannesburg and Bloemfontein, all described in
my historical sketch, were undertaken with the definite purpose
of providing the basis for such statistical studies.
The Lick Observatory survey was completed in 1915, and the
results obtained for the sky area north of the equator will now be
presented. The data consist of all known double stars as
bright as 9.0 BD magnitude* which fall within the distance
limits set by the following working definition of a double star
* BD magnitudes were taken because photometric magnitudes to the
limit 9.0 were not available.
257
258 THE BINARY STARS
proposed by me in 1911:
(1) Two stars shall be considered to constitute a double star when
the apparent distance between them falls within the following limits:
1" if the combined magnitude of the components is fainter than 11.0
3" if the combined magnitude of the components is fainter than
9.0 BD.
5" if the combined magnitude of the components lies between 6.0
and 9.0 BD.
10" if the combined magnitude of the components lies between 4.0
and 6.0 BD.
20" if the combined magnitude of the components lies between 2.0
and 4.0 BD.
40" if the combined magnitude of the components is brighter than
2.0 BD.
(2) Pairs which exceed these limits shall be entitled to the name
double star only when it has been shown (a) that orbital motion exists;
(6) that the two components have a well defined common proper motion,
or proper motions of the 61 Cygni type; (c) that the parallax is decidedly
greater than the average for stars of corresponding magnitude, t
In all, there are 5,400 pairs, more than half of which were
discovered in the course of the survey (766 by Hussey, 2,057 by
* [See pages 35 and 268. Aitken's working definition (1911) is:
log P = 2.5  0.2m
which gives 5" for 9.0 magnitude.
Also, see ADS, p. IX, separation limits for ADS
log p = 2.8  0.2m
pp. m
P
App. m
P
1.0
400"
7.0
25"
2.0
250"
8.0
16"
3.0
160"
9.0
10"
4.0
100"
10.0
6"
5.0
63"
11.0
4"
6.0
40"
12.0
275— J.T.K.]
t The definition, with correspondence relating to it, will be found in the
Astronomische Nachrichten (188, 281, 1911). Comstock and E. C. Pickering
there suggest limits based upon the apparent magnitude, the former using
the formula s = c($j) m , the latter, the formula, log s = c — 0.2m, where s
is the distance in seconds of arc between the components, m, the apparent
magnitude, and c an arbitrary constant. If the values of c in the two for
mulae are so chosen as to give the limit 5'0 for stars of magnitude 6.0, the
formulae will give the limits 0'75 and 1 "25, respectively, for stars of 9.0 mag
nitude. From the theoretical point of view either formula gives more
logical limits than the ones in my definition, but there were practical con
siderations, fully stated in the article referred to, which led to the adoption
of the latter.
DATA RELATING TO STARS IN THE NORTHERN SKY 259
Aitken). A given system is counted only once though it may
have three, or even four or more, components. In the multiple
systems the closer pair is, in general, the one counted, but in a
few cases in which the close pair is very faint, the principal
bright pair is taken.
The first question to consider is whether the data are homo
geneous, for it is obvious that they can make no claim to be
exhaustive. However carefully an observer may work, some
pairs which he might discover with a given telescope will surely
escape him. His eye may be fatigued, unnoticed haze or
momentary bad seeing may blur out a faint companion star,
or it may chance that at the date of examination the two com
ponents are so nearly in conjunction as to be below the resolv
ing power of the telescope. The number of known double
stars can only be regarded as the lower limit to the number
which might be discovered. Homogeneity was earnestly
sought for, care being taken to work only when in good phys
ical condition and when the seeing was good, the practical test
being the power to recognize very close and difficult pairs at
a glance. But variations in the conditions are inevitable when
the working program requires years for its execution and doubt
less such variations have affected the present results. Careful
comparison, however, shows no discernible difference in the
thoroughness of the work done at different seasons of the year
or in different parts of the sky, and it may fairly be said that
the results of the survey represent the capacity of the com
bination of telescope and observer under average good atmos
pheric conditions at Mount Hamilton. If the work had all
been done with the 36in. refractor the resulting data might
be considered quite homogeneous. Unfortunately, a con
siderable part of it, including practically the entire area north
of +60° declination, was done with the 12in. telescope, and it
becomes necessary to consider the relative efficiency of the
two instruments.
I have applied two tests: first, the comparison of the most
difficult pairs discovered with each instrument; second, the
reexamination with the 36in. of some 1,200 stars previously
examined with the 12in. telescope. I find that, under the
usual observing conditions, a pair with nearly equal components
separated by only 0"15, or a companion star as faint as 14.5 mag
nitude and not less than 1"5 from its primary is practically
260
THE BINARY STARS
certain of detection* with the 36in.; with the 12in., the cor
responding limits in the two cases are 0"25 and 13 to 13.5 mag
nitude. Twelve new double stars were added by the reexamina
tion of the 1,200 stars. From these tests, taking into account
the proportion of the whole work done with the 12in. telescope,
I conclude that about 250 pairs would have been added if the
entire northern sky had been surveyed with the 36in. Since
all or nearly all of the brighter stars had been examined repeatedly
by other observers using powerful telescopes, it is fair to assume
that comparatively few of these undiscovered pairs are brighter
than 7.0 BD magnitude.
According to Seeliger's count of the BD stars there are 100,979
as bright as 9.0 magnitude in the northern hemisphere. Of
these, 5,400, or 1 in 18.7 on the average, have actually been
found to be double within the limits set above. If we add
only 200 pairs, the ratio becomes 1 : 18.03. A definite answer is
Table 1. — The Distribution of Double Stars in Right Ascension and
Declination
R.A.
Decl.
0>
l h
2 h 
3 h
4 h 
5 h
6 h 
7 h
8 h 
9"
ltf 1 
ll h
12 h 
13 h
14 h 
15 h
16 h 
17 h
18 h 
ltf 1
20 h 
21 h
22 h 
23 h
0°9°
6.3
5.4
5.2
6.0
6.7
6.2
4.6
6.3
5.4
5.6
5.5
5.0
6.2
5.0
7.4
6.4
5.9
5.9
6.0
6.0
5.5
4.8
4.3
4.5
4.8
4.3
3.1
ll h )
5.9
5.5
6.4
4.8
6.2
3.6
4.4
2.6
3.3
5.5
4.8
5.2
5.4
3.2
4.4
2.6
2.8
(12 h 
6.4
6.1
4.8
4.1
5.6
6.0
4.1
2.2
17 h )
5.4
4.5
5.4
5.0
5.5
4.6
3.6
3.4
3.4
6.0
6.0
6.2
5.3
5.4
4.9
5.2
5.2
4.9
1019
2029
6.2
5.1
4.4
4.5
5.2
2.9
3.5
(6 h 
5.2
4.2
3039
5.6
4.2
4.4
4.1
4.9
(18 h 
4.8
4049
5.0
8.0
7.1
4.2
3.7
4.9
5059
6069
6.8
5.3
5.0
23 h )
4.9
6.8
7079
8089
5.S
(0 h 
► 3.8
5 h )
6.5
3.9
The figures give the percentages of double stars among stars to 9.0 BD magnitude; the
average percentage for the whole northern sky is 5.35.
thus given to my first question: At least one in every 18, on the
average, of the stars in the northern half of the sky which are as
bright as 9.0 BD magnitude is a close double star visible with the
36t'n. refractor.f There is no reason to doubt that the ratio is
* Unless the primary is brighter than 7.0 magnitude.
t [Research and observation since 1935 lead to the conclusion that at least
half the stellar population consists, not of single stars like our sun, but of
members of systems; that is, practically one out of every two stars is a binary
or multiple system. Also, in spite of the high frequency of spectroscopic
binaries among bright stars, W. H. van den Bos assures us that in space the
spectroscopic binary is a rare exception, the visual binary the rule.— J.T.K.]
DATA RELATING TO STARS IN THE NORTHERN SKY 261
equally high in the southern half of the sky. In fact, preliminary
counts made by van den Bos lead him to estimate the ratio in
that hemisphere as high as one in 17.
Table 1 exhibits the distribution of the 5,400 double stars in
right ascension and declination as compared with the distribu
tion of the BD stars to 9.0 magnitude, the figures giving the
percentage of double stars in each area. There are obvious
irregularities in the table but no evidence of systematic differ
ences that can be regarded as seasonal effects. The percentages
are as high in the sky areas surveyed in winter as in those sur
veyed in summer. There is a falling off in the percentage in
the high declinations, especially in the regions well removed
from the Milky Way, which is doubtless due in part to the
fact that the area north of 60° was almost entirely surveyed
with the 12in. telescope. The broken line in the table represents
very roughly the position of the central line of the Milky Way,
and it will be noted that the percentages near this line are, in
general, above the average.
The distribution with respect to the plane of the Milky Way
is more clearly brought out when the stars are tabulated accord
ing to galactic latitude. This has been done in Tables 2 and 3,
in the former of which the stars are divided into classes accord
ing to magnitude and the latitudes into zones each 20° wide,
Table 2. — The Distribution of Double Stabs by Magnitude Classes
and Zones op Galactic Latitude
r/^^v^ Mag.
to 6.5
6.67.0
7.17.5
7.68.0
8.18.5
8.69.0
Total
Zone ^n.
I
19
13
14
29
40
84
199
II
43
28
50
68
114
193
496
III
60
43
56
79
148
254
640
IV
96
54
81
132
232
401
996
V
121
88
133
249
376
653
1,620
VI
84
51
81
134
221
395
966
VII
28
23
18
54
90
154
367
VIII
7
6
5
12
31
55
116
Total
458
306
438
757
1,252
2,189
5,400
beginning at the north galactic pole. Zone V therefore includes
the area from +10° to —10° galactic latitude, and zone IX,
which ends at the south galactic pole and lies entirely below
the equator, is not represented. As was to be expected, the
numbers in every column of this table are largest in zone V
262
THE BINARY STARS
and fall to minima in zones I and VIII. The question is whether
this condensation toward the Milky Way is greater than that
of all the stars. Table 3 provides the answer. Since the
zones are not of equal area, and since only the first one lies
wholly in the northern hemisphere, the fairest comparison
is that afforded by the relative densities per square degree of
double stars and of all stars of the corresponding magnitudes.
The double star densities were determined by dividing the
figures in Table 2 by the number of square degrees in each
zone area; the figures were then reduced to a common standard
by making the density in each column unity in zone V, the
Milky Way zone. Seeliger has published corresponding data
Table 3. — Density op Double Stabs by Magnitude Classes and
Galactic Latitude Compared with the Density op BD Stabs to
9.0 Magnitude (Afteb Seeligeb)
n. Mag.
to 6.5
6.67.0
7.17.5
7.68.0
8.18.6
8.69.0
Zone >v
BD
D.S.
BD
D.S.
BD
D.S.
BD
D.S.
BD
D.S.
BD
D.S.
I
II
III
IV
V
VI
VII
VIII
0.551
0.572
0.639
0.790
1.000
0.912
0.572
0.428
0.395
0.456
0.480
0.789
1.000
0.822
0.395
0.307
0.431
0.445
0.554
0.689
1.000
0.787
0.427
0.315
0.374
0.410
0.474
0.610
1.000
0.687
0.446
0.361
0.518
0.497
0.599
0.765
1.000
0.842
0.467
0.352
0.266
0.484
0.407
0.606
1.000
0.721
0.231
0.199
0.404
0.424
0.509
0.730
1.000
0.772
0.480
0.373
0.304
0.351
0.307
0.529
1.000
0.639
0.370
0.255
0.419
0.441
0.512
0.720
1.000
0.799
0.521
0.462
0.261
0.390
0.380
0.614
1.000
0.694
0.407
0.435
0.382
0.404
0.484
0.728
1.000
0.789
0.527
0.527
0.325
0.380
0.377
0.613
1.000
0.716
0.401
0.445
for all of the BD stars and the two sets of values are entered
in Table 3 in the columns D.S. and BD, respectively. It is
clear that the density curves of double stars rise to sharper
maxima in the zone V than the corresponding curves of stars
in genera] do.
This fact is exhibited in a more striking manner if we tabulate,
as in Table 4, the percentages of double stars in five areas, the
Milky Way zone, the 20° zone on either side of it and the areas
north of +30° and south of —30° galactic latitude.
The increased percentage in zone V must be accepted as
real. Table 3 shows that stars of all magnitude classes par
ticipate in it, and an examination of my charts leads to the
conclusion that it cannot be an observing effect, for some areas
of all galactic latitudes were examined in summer, others in
winter; the area north of +60° declination, examined almost
exclusively with the 12in., extends from —3° to +27° galactic
DATA RELATING TO STARS IN THE NORTHERN SKY 263
latitude and the areas of high galactic latitude, both north and
south, were examined mainly with the 36in. refractor. It
appears, therefore, that among stars as bright as 9.0 BD magni
tude close visual double stars are relatively more numerous in
low than in high galactic latitudes.
Table 4. — Percentages of Double Stars
Galactic latitude
BD stars to 9.0
Double stars
Percentage of
double stars
+90° to +30°
+30 +10
+10 10
10 30
30 70
26,948
19,355
26,477
17,831
10,368
1,335
996
1,620
966
483
4.95
5.15
6.13
5.13
4.66
This apparent concentration of double stars toward the galactic
plane is certainly to be explained, in part, by the far greater
extension of the stellar system in that plane than in the direction
perpendicular to it. Possibly this is the full explanation,
perhaps the observed increase in double star density is entirely
a perspective effect; but in that event it would seem that in
zone V, the galactic zone, we might expect a relatively higher
percentage of very close pairs than of pairs of greater separation.
Table 5, however, in which the 5,400 pairs are grouped according
to galactic latitude and angular separation, shows that the
percentage increase toward zone V is substantially the same in
all the angular distance categories up to 5" 00.
Let us consider next the relation between the angular sep
aration and magnitude. This is shown in Table 6 where the
pairs are arranged with these qualities as arguments. The
Table 5. — The Distribution of Double Stars in Galactic Latitude
by Distance Classes
\Dist.
o'.'oo
0751
1701
1751
2"01
3701
4701
5701
\
0750
1700
1750
2700
3700
4700
5700
and over
Zone\
No.
%
No.
%
No.
%
No.
%
No.
%
No.
%
No.
%
No.
%
I
41
11
31
10
25
14
24
16
39
17
15
9
14
10
10
20
II
101
26
92
30
56
30
56
38
61
26
65
40
48
33
17
34
III
139
36
99
32
90
49
65
44
101
43
61
37
53
36
32
64
IV
225
58
175
57
131
71
104
70
133
57
112
68
74
51
42
84
V
388
100
306
100
184
100
149
100
233
100
164
100
146
100
50
100
VI
247
64
152
50
119
65
105
70
144
62
101
62
72
49
26
52
VII
82
21
77
25
52
28
36
24
42
18
32
20
29
20
17
34
VIII
33
8
18
6
14
8
12
8
12
5
9
6
12
8
6
12
264
THE BINARY STARS
Table 6. — The Distribution of Double Stabs bt Angulab
Distance and Magnitude
\Dist.
to
0'51
o'oo
l"01
2"01
3'01
4''01
5"01
and
over
Mag.\
0"50
l'OO
l'OO
2"00
3'00
4"00
5"00
g 6.5
75
63
138
83
62
41
31
99
6.67.0
82
52
134
59
42
40
21
14
7.17.5
103
67
170
99
64
48
31
29
7.68.0
178
132
310
164
107
85
63
26
8.18.5
310
223
533
285
173
128
111
21
8.69.0
508
413
921
532
317
217
191
11
Totals
1,256
950
2,206
1,222
765
559
448
200
Percentages
^ 6.5
6.67.0
7.17.5
7.68.0
8.18.5
8.69.0
Totals
39
45
41
42
43
42
42
23
20
24
22
23
25
23
17
14
16
15
14
14
15
12
14
12
12
11
10
11
sums of the numbers in the first two columns of the table are
entered in the third, thence the numbers are given for uniform
steps in angular distance to the final column. Every line of
columns three to seven exhibits a marked increase in the number
of pairs as the angular distance diminishes.
This is still more apparent when the figures in these five col
umns are expressed as percentages of the total number of pairs
under 5"0 separation in each magnitude class. If we may
assume that the stars of a given magnitude class, for example,
from 8.6 to 9.0, are, on the average, at the same distance from
us, then this observed increase in the number of pairs as the
angular distance diminishes is not a mere perspective effect
but represents a real increase in the number of pairs as the angular
distance diminishes.
The table also indicates that the Lick Observatory survey
was as thorough for the fainter stars (to 9.0 BD magnitude)
as for the brighter ones, for the percentages in each column
in the lower division of the table are not far from uniform and
there is a fair agreement between the ratios of the figures in each
line of the first two columns of the upper half of the table. This
was to be expected, for the pairs most likely to be missed in a
DATA RELATING TO STARS IN THE NORTHERN SKY 265
survey are those in which the difference in magnitude of the
two components is large and bright pairs of that character are
at least as difficult objects as pairs in which the primary is
of 9.0 magnitude.* When the two components are of equal
brightness, on the other hand, a 9.0 magnitude pair is but little
more difficult than one of 6.0 magnitude, unless the angular
distance is below 0"15; in fact, I find that 123 of the 379 pairs
with angular distance less than, or equal to, 0"25, and 385 of
the 877 pairs with angular distances between 0"26 and 0"50,
discovered in that survey are of the BD magnitude class 8.69.0.
These statements are of significance also in connection with the
figures entered in Table 7, which shows the percentage of double
stars of each BD magnitude class.
Table 7. — Percentage of Double Stabs by Magnitude Classes
Magnitude
to 6.5
6.67.0
7.17.5
7.68.0
8.18.5
8.69.0
BD stars
4,120
3,887
6,054
11,168
22,898
52,852
Double stars
458
306
438
758
1,251
2,189
Percentage of
double stars
11.1
7.9
7.2
6.8
5.5
4.1
The drop in percentage as we pass from the brighter to the
fainter magnitudes cannot be attributed to lack of completeness
in the data, for it follows from what has been said above, that the
pairs overlooked in the survey are quite as likely to belong to
the brighter magnitude classes as to the fainter ones; but we must
not forget that this table, as well as the preceding ones, is based
upon the apparent magnitudes only; whether the observed
progression holds also for the absolute magnitudes we shall
not know until we have more knowledge of the parallaxes of
the stars involved. The very high percentage of spectroscopic
binaries, among stars as bright as or brighter than 5.5 apparent
magnitude, and particularly among stars of classes A and B,
may possibly be significant in this connection, though we cannot
as yet say definitely that the percentage is not as great among the
fainter stars.
It will be of interest to inquire whether the results for the
relative numbers and apparent distribution of the visual binary
* This holds true unless Am exceeds six magnitudes.
266
THE BINARY STARS
stars given by the survey just reviewed find any support from
the data given in the New General Catalogue of Double Stars
within 120° of the North Pole, or from other available data.
In 1932, I made a single count of the double stars in the new
catalogue which are included within the limits of the curve
defined by the formula log p = 2.5 — 0.2m. This formula
gives 5''0 as the angular separation corresponding to a pair of
9.0 magnitude. I found 12,708 such pairs. Of these, 4,761 pairs
are fainter than 9.0 magnitude and must be omitted in our
present inquiry because no adequate double star survey has
yet been made of the stars below 9.0. Further, it must be said
that the survey is not complete even to 9.0 for the entire sky
down to —30° declination. Making a very generous estimate
of the number of such stars actually examined, I again find that
at least one in every 18 is a close double star within the resolving
power of good modern telescopes.
Table 8 shows the distribution of these 7,947 pairs by magni
tude and angular separation, the data being given in more
condensed form than was adopted in the earlier tables to minimize
the effect of errors in the rapid count of the different categories.
Table 8
^\. Ang. sep.
<0''51
0'51
l''00
l"01
2''00
2''01
5"00
5'01
Curve
Total
< 6.00
6.017.00
7.018.00
8.019.00
66
136
305
879
69
98
244
793
63
134
346
1,035
156
214
522
1,564
317
302
341
363
671
884
1,758
4,634
Totals
Percentages
1,386
17.4
1,204
15.2
1,578
19.9
2,456
30.9
1,323
16.6
7,947
100.0
The number of pairs under 0''51 in angular distance exceeds
the number with distances between 0"51 and l''OO; the number
under l''OO is about onethird, the number wider than 5"0 is
but onesixth of the whole. These figures support the earlier
finding that the number of double stars increases as the angular
distance diminishes.
The fact that the tables presented above are based upon the
apparent magnitudes, the observers' estimates (often very
DATA RELATING TO STARS IN THE NORTHERN SKY 267
inexact) of the difference in magnitude of the two components
of a pair, and upon measures of the angular distances made
at many different epochs and with telescopes both large and
small, is to be emphasized. If accurate photometric measures
of magnitude (and of Aw) were available, and fairly reliable
measures of all angular distances, it would be desirable to set up
tables with the arguments log distance, m, and Aw. Such
tables would give a far better picture of the actual distribution
of the double stars.
It is well known that four of the ten stars nearest to the Sun
are visual binaries: a Centauri, Sirius, Procyon, and 61 Cygni.
This, no doubt, is an exceptional percentage, but it is of interest
to quote some figures sent me by van Maanen who has recently
made a study of the nearest stars.* Within a distance of
5 parsecs (approximately 16 light years), he finds that 8 double
star systems (with 18 components) and 21 single stars (including
the Sun) are known; in the volume of space between the limits
5 and 10 parsecs, 16 double or multiple systems (38 components;
and 62 single stars, and in the volume between 10 and 20 parsecs,
91 double or multiple systems (197 components) and 281 single
stars. In all, 617 individual stars (including the Sun) are known
to us in the volume of space within a radius 'of 20 parsecs. Of
these, 364 are single stars; the remaining 253 are components of
115 double or multiple systems. Counting each such system
as a unit, then, one star in four is a visual double or multiple.
Van Maanen proceeds to show that there are probably about
2,000 individual stars in this volume of space that have not
yet been recognized as inhabitants. The chances are that the
percentage of components of double star systems is not so great
among these fainter stars as among those already known.
Counts like these cannot be compared directly with the results
given by our double star survey, but they nevertheless indicate
that our estimate of the frequency of double stars is a very
conservative one.
The spectral classes of the stars have been ignored in the
preceding tables. It will be of interest now to take them into
account, as far as possible. In 1917, through the courtesy of
the Director, the late E. C. Pickering, and of Dr. Annie J. Cannon,
of the Harvard College Observatory, I had the privilege of
comparing my list of 5,400 double stars with the great card
* Publ. A.S.P. 46, 247, 1933.
268
THE BINARY STARS
catalogue of stellar spectra later published as the Henry Draper
Catalogue. This comparison provided the spectral classification
of 3,919 of the 5,400 pairs. Of the remaining 1,481, only 15
are as bright as 8.0 (BD), 218 lie between 8.1 and 8.5, and
1,248 between 8.6 and 9.0. The published volumes of the
Henry Draper Catalogue were available when I prepared the
copy for my New General Catalogue of Double Stars within 120°
of the North Pole (briefly the ADS), and I utilized them to enter
all known spectra of the double stars I catalogued. I have made
a single count of these, taking the spectrum of the primary star
only, in cases where the spectra of additional components are
entered, and find the total to be 9,190.
The pairs in the ADS, include all double stars to —30° declina
tion, falling within the very generous distance limits set by the
curve log p = 2.8 — 0.2m, which gives the angular distance
10"0 for a 9.0 (BD magnitude) pair.
Table 9 shows the distribution according to spectral class of
the 9,190 ADS pairs with known spectra, of the 3,919 pairs in
the northern half of the sky which fall within the much narrower
limits set by my "working definition," and of the 222,570 stars
in the Henry Draper Catalogue, as given in H.C.O. Circular 226.
Table 9
Numbers
Visual pairs in n. hem. ,
Visual pairs in ADS . . .
Stars in HDC
Ratio ADS: HDC.
B
A
F
G
K
M
157
1,251
532
1,093
837
49
268
2,910
1,220
2,461
2,126
205
3,567
64,259
21 , 120
46,552
73,208
13,864
1:13
1:22
1:17
1:19
1:34
1:68
All
3,919
9,190
222,570
Percentages
Visual pairs in n. hem.
Visual pairs in ADS . .
Stars in HDC
4.0
31.9
13.6
27.9
21.4
1.2
2.9
31.7
13.3
26.8
23.1
2 2
1.6
28.9
9.5
20.9
32.9
6.2
100.0
100.0
100.0
Table 10, in similar manner, shows the distribution of the
spectroscopic binary stars in Moore's Third Catalogue (Table I),
omitting the Cepheid variables and other stars whose variation
in radial velocity does not satisfy Otto Struve's criterion. Since
these binaries are practically all bright stars — nearly all brighter
DATA RELATING TO STARS IN THE NORTHERN SKY 269
than 5.5 visual magnitude — their distribution in spectral class
is compared with that of the HDC stars as bright as 6.25 magni
tude (H.C.O. Circ. 226).
Table 10
Numbers
B
A
F
G
K
M
All
Spec. Bin.
HDC Stars to 6.25
Ratio Spec. Bin. to HDC
227
719
1:3
360
2,018
1:6
104
680
1:7
115
656
1:6
149
1,984
1:13
26
538
1:21
981
6,595
Percentages
Spec. Bin.
HDC Stars to 6.25
23.1
10.9
36.7
30.6
10.6
10.3
11.7
9.9
15.2
30.1
2.7
8.2
100.0
100.0
The upper part of each table records the actual numbers of
stars counted, the lower part their percentage distribution among
the spectral classes. It is evident that the count of the 3,919
visual parts made in 1917 gave a good representation of the
apparent distribution of the visual binary stars among the
different spectral classes, for the count of the 9,190 pairs exhibits
very closely the same percentage distribution.
It appears from the frequency ratios in Table 9 that visual
binary stars are relatively more numerous among stars of
classes B, F, and G, than among the stars of the other three
classes, the small ratios for classes K and M being specially
striking. From Table 10 it is equally clear that among the stars
as bright as 6.25 magnitude, spectroscopic binaries are most
numerous among stars of Class B and least numerous among
stars of Class M. The strong contrast between the frequency
ratios for classes B and M in both tables is perhaps their most
striking feature. It may be of interest to add that my count
shows that the visual binaries of classes B and A in the ADS
are strongly concentrated toward the galactic plane, whereas
those of classes F and M are quite uniformly distributed over the
sky, and those of classes G and K are more frequent among stars
of high than of low galactic latitude.
Such general statistics are of interest and have also a certain
degree of significance, as have also the statistical relations
270 THE BINARY STARS
between the magnitudes, spectral classes and colors of the
components of the visual binary stars. The relation between
the colors of the components of double stars and their difference
in magnitude was recognized by Struve and every observer
since his time has noted that fact that when the two components
are about equally bright they are almost without exception of
the same or nearly the same color, and that the color contrast
increases with the difference in magnitude of the components.
Professor Louis Bell argued that this is a subjective effect
since the fainter star is generally the bluer one. Doubtless this
subjective effect is often present but it is by no means the sole
cause. There are real and very striking differences in the
spectral classes of the components of double stars and these
are definitely correlated to the color differences and also to the
differences in magnitude. The absolute magnitude of the
primary also enters as a factor. Thus Lau,* in two papers
written as early as 1917 and 1918, respectively, found that the
companions to giant stars are bluer, the companions to dwarf
stars redder than their primaries.
Several writers have investigated the relations between
magnitude and spectral class, in the components of double stars,
among them Dr. F. C. Leonard, f His thorough analysis of
the data for 238 pairs clearly showed (1) that when the two
components are of equal or nearly equal magnitude, they differ
little in spectral class, except in the case of a few giant stars
like 7 Circini; (2) that when the primary is a giant of spectral
class F0 or later, the companion belongs to an earlier spectral
class; and (3) that when the primary is a dwarf star, or a giant
star of spectral class earlier than F0, that is, when the primary is a
star of the main sequence, the companion belongs to a later
spectral class. These results prove that the color relationships
observed by Lau correspond to actual differences of spectral
class. Since they are also in harmony with the massluminosity
relationship, they indicate that the components of double stars
are normal stars, having the same properties as ordinary single
stars of corresponding mass and magnitude.
* A.N. 205, 29, 1917; 208, 179, 1918.
t Leonard, Lick Obs. Bull. 10, 169, 1923. See also Peter Doig's paper
(Mon. Not. 82, 372, 1922) and G. Shajn's (Bull. Poulkova Obs. 10, 276,
1925). Their conclusions are similar to Leonard's.
DATA RELATING TO STARS IN THE NORTHERN SKY 271
As has already been noted, one of the weakest points in these
statistical investigations aside from the unavoidable incom
pleteness of the data, is that the apparent magnitudes of the
components of double stars are so unreliable. What is needed
is the accurate determination of Am (and therefore of difference
in absolute magnitude) of all pairs to a given magnitude limit
(for the primary star). Some years ago, Dr. G. P. Kuiper,
with the aid and encouragement of Prof. Ejnar Hertzsprung,
began to make such a determination, using suitable wire gratings
over the 10in. refractor at Leiden, to reduce the firstorder
spectra of the primary stars to a brightness within onehalf a
magnitude of the image of the secondary star. He is following
out this program at the Lick Observatory at the present time,
with the object of including in his observations every star as
bright as 6.5 visual magnitude on the Harvard scale, that has a
companion within 30". He is at the same time determining the
spectral classes of the components, using a slitless spectrograph
attached to the Crossley reflector. When completed, this
program will afford a sound observational basis for a study
of the relationships between Aw and the difference of spectral
class in these bright pairs.
The various results as to the number and distribution of the
binary stars given above are all, as I have said, of interest and of
significance, though they rest, admittedly, upon data neither
homogeneous nor complete. Some will be confirmed when
additional data become available, others may have to be aban
doned. One of the most significant findings is perhaps, that the
components of the binary stars are normal stars, resembling in
all essentials the single stars of corresponding magnitude and
spectral class.
Among other investigations that lead to this same conclusion
we may refer to Oort's* "A Comparison of the Average Velocity
of Binaries with That of Single Stars," in which he finds no
indication of any difference between the two; and Wallinquist'sf
"The Solar Motion as Derived from the Radial Velocities of the
Visual Binary Stars." His value for the position of the solar
apex, derived from a discussion of 536 systems, mainly with
angular distance between components under 5", agrees closely
with the generally accepted value.
* J. H. Oort, A.J. 35, 141, 1924.
t A. Wallinquist, Ann. Bosscha Sterremoacht 4, 21, 1929.
272 THE BINARY STARS
References
Many papers have been written in the past fifteen years on questions
related to those discussed in this chapter. In addition to those already
referred to in the footnotes, a few representative papers are listed here.
Jackson, J.: Mon. Not. R.A.S. 83, 4, 1922.
Lundmark, K., and W. J. Luyten: On the Relation between Absolute
Magnitude and Spectral class as Derived from Observations of Double
Stars, A.J. 35, 93, 1923.
Luyten, W. J.: On Some Statistical Properties of Double Stars in Space,
Proc. Nat. Acad. Set. 16, 252, 1930.
Gyllenberg, W. : The Binary Stars and the Stream Motions, Med. Lunds.
Obs. Ser. 1, No. 132, 1932.
Also a series of papers by E. A. Kreiken in the Monthly Notices.
CHAPTER XI
THE ORIGIN OF THE BINARY STARS
The problem of the origin of the binary stars may be con
sidered from two quite different points of view. We may ques
tion by what conceivable process or combination of processes
a single star or, alternately, a primal nebulous mass, can develop
into a binary system of any kind whatever; or, we may pass in
review the whole vast series of spectroscopic and visual double
and multiple systems and ask what theory is competent to
account not only for the origin of any one binary system but
also for the great variety in content, form, and dimensions the
known systems actually exhibit. Obviously, the second point
of view makes the severer demand upon any theory that may be
propounded, whether we regard all binaries as objects of a single
genus, as I am disposed to do, or separate the very close, short
period spectroscopic binaries from those of longer period and
from the visual binaries.
We may summarize briefly the more important conclusions
that have been reached in the discussion of the observations of
the visual double stars in the northern half of the sky and from
the study of the known orbits of the visual and spectroscopic
binary systems presented in the three preceding chapters.
1. A large proportion of the stars are binary systems. On
the average, at least one in 18 of those as bright as 9.0 magnitude
is a binary visible in our telescopes and coming within the
limits set by the "working definition" of a double star; at least
one in every three or four of those as bright as 5.5 magnitude is a
binary revealed by the spectrograph. These are minimum
values. Both visual and spectroscopic binaries within these
magnitude limits remain to be discovered, our knowledge of
the fainter stars is still very incomplete, and the angular distance
limits set by the "working definition" are necessarily arbitrary.
Unquestionably, many double stars with greater angular dis
tances are binary systems. On the evidence before us, we may
safely say that onethird, and probably twofifths of the stars
are binary or multiple systems.
273
274 THE BINARY STARS
2. A considerable percentage of these systems have three or
more components. It is well within the truth to say that one
in 20 of the known visual binaries has at least one additional
member either visible or made known by the spectrograph, and
many systems are quadruple or even more complex. Many
of the purely spectroscopic systems are also triple or multiple.
It is a fact of undoubted significance that, as a rule, triple
systems, whether visual or spectroscopic, consist of a close binary
pair and a companion relatively distant.
3. The masses of the spectroscopic binaries of Class B are
decidedly greater than those of the spectroscopic binaries of
later classes, and, among the doubleline binaries (i.e., those in
which the spectra of both components are visible) there is some
evidence of a progressive decrease in the average mass values
with advancing spectral class. Evidence of the same kind,
though less marked, exists for the visual binaries for which
both orbits and parallax values are available, for the average
masses of the visual binaries of classes A, F, and G agree very
well with those of the spectroscopic binaries of the corresponding
classes. It is to be noted, however, that doubleline spectro
scopic binaries and visual binaries with known orbits and
parallaxes are necessarily selected systems.
4. The rule, both in visual and in spectroscopic binaries, is
that the fainter component is the less massive one, but the mass
ratio is rarely less than onehalf and the average, as far as can be
determined from the available data, is about threefourths.
5. Spectroscopic binaries are relatively more numerous among
stars of classes B and A, and visual binaries among stars of
classes B, F, and G, than among stars of the other spectral
classes. The small number of systems of either type among
stars of Class M is specially striking.
6. When the primary star of a visual binary is a giant of
spectral class later than F0, the companion belongs to an earlier
spectral class; when the primary is a star of the main sequence
(i.e., a giant of spectral class earlier than F0, or a dwarf) the
companion belongs to a later spectral class, the difference in
spectral class, in both cases, increasing with the difference in
magnitude of the two components.
7. There is a close correlation between the length of period,
or size of system, and the degree of ellipticity in the orbit. The
visual binaries, with periods to be reckoned in years or even in
THE ORIGIN OF THE BINARY STARS 275
centuries, have an average eccentricity slightly above 0.5; the
spectroscopic binaries, with periods to be reckoned, for the most
part, in days or even in fractions of a day, have an average
eccentricity of less than 0.2, and in each class the average eccen
tricity increases with the average length of period. We have,
apparently, one unbroken progression or series of orbits from
systems in which the two components revolve in a fraction of a
day in circular orbits and practically in surface contact, to
systems in which the components, separated by one, two, or
many hundreds of times the distance from the Earth to the
Sun, revolve in highly elliptic orbits in periods of hundreds and
even of thousands of years. The scatter of the values of e in
any given period group is large, but, statistically, we may regard
the correlation as securely established.
These facts and relations, as well as others less conclusively
demonstrated as yet, must all be taken into account when we
look at the question of the origin of the binary stars from the
second point of view indicated in the opening paragraph of this
chapter. We have now to see whether any of the theories so
far advanced are satisfactory from this point of view. At least
three theories have been developed that merit consideration.
The capture theory, apparently first advanced by Dr. G. John
ston Stoney, in 1867, is based upon the hypothesis that two
stars, originally independent, might approach each other under
such conditions that each would be swerved from its path and
forced to revolve with the other about a common center of
gravity. This theory, in its original form, has been completely
abandoned, but the consequences of the near approach of two
stars have been the subject of extensive discussion in more
recent years. Chamberlin and Moulton, for example, have
argued that, under appropriate initial conditions, such an
approach might result in a system of planets like our own. This
is probably the best theory that has been developed to account
for the origin of the solar system, and if it may be accepted, then
we may agree that the close approach of two stars, under some
what different conditions, might result in a planetary system
in which the disparity in mass between the central star and its
largest planet would be far less than it is between the Sun and
Jupiter. That is the assumption MacMillan* makes in arguing
* Science 62, 63, 96, 121, 1925.
276 THE BINARY STARS
that, in certain cases, a planetary system might ultimately
develop into a binary star system.
Assume, for example, that, as the solar system traverses space,
the Sun and the planets grow in mass from the infall of atoms,
molecules or larger particles, sometimes very slowly, at others
(as the system passes through a region filled with nebulous
matter) more rapidly, and that, on the whole, the Sun's gain by
this process is offset by its loss of mass through radiation. Then
the mass of Jupiter, to consider only the largest planet, would
steadily increase relatively to that of the Sun and the distance
between the two bodies would decrease. If, in this manner
Jupiter were to acquire sufficient mass to become a dwarf red
star while the Sun just held its own in mass, we should have a
double star system, the other planets being absorbed, ultimately,
either by Jupiter or by the Sun. MacMillan describes the
process in more detail, but in a recent personal letter says that
Jupiter could not become equal to the Sun in mass before it
merged with it. Indeed it is doubtful if it could exceed one
tenth the mass of the Sun without having the two bodies drawn
together.
It is hard, therefore, to see a double star future for the solar
system, for we have found that even in the closest spectroscopic
binaries the mass ratio rarely falls below onehalf. But even
if we grant that a binary star might develop, in the course
of eons, from a planetary system in which the original mass
distribution was more favorable, the process, at best, could
account for only an occasional spectroscopic binary with quite
unequal components. Once formed, there would, it is true, be a
tendency toward equality in mass of the two components, as
time went on, for the rate of radiation from a star is proportional
to a power higher than the first,* but we have already noted
MacMillan's conclusion that in a system so formed the two
stars are likely to fall together when the mass ratio approaches
unity. If we assume this danger to be averted through tidal
interaction and loss of mass by both bodies through radiation,
then, statistically, the masses of the visual binaries with periods
measured in decades or in centuries should be smaller than those
of the spectroscopic binaries, with periods measured in days.
This, however, we have shown not to be the case (Tables 10 and
* Jeans, M on. Nat. R.A.S. 85, 209, 1925.
THE ORIGIN OF THE BINARY STARS 277
11, Chap. VIII). We must, I fear, decide that the theory does
not meet the requirements.
The fission theory which we have next to examine, assumes
that a star in its primal nebulous stage, or possibly at a later
one, divides under the stress of its own gravitation, radiation
pressure, and rotational forces or under the strain of some
external disrupting force or forces.
The behavior of a rotating, homogeneous, incompressible,
fluid mass, in equilibrium and free from external disturbance,
was investigated by Maclaurin and Jacobi about a century
ago and later by Poincare\ G. H. Darwin, Liapounoff, Jeans,
and others. It was found possible, under certain assumptions,
to follow, mathematically, the transformations of figure as
the rotating mass contracts under its own gravitation and heat
radiation from the initial sphere through a succession of spheroids
and ellipsoids until a pearshaped figure is reached. It seems
probable, though it could not be demonstrated mathematically,
that fission into two masses would follow, the masses revolving
at first in surface contact and in circular orbits.
The stars and, in all probability, the antecedent nebulae,
are neither homogeneous nor incompressible, but it has been
argued, first, I believe, by See, and later by Darwin and Jeans,
that a nebula (or even a star in its earliest stage) might none
the less pass through a similar series of changes and ultimately
form a stable double star system. Once formed, the forces of
tidal interaction and of the disturbances ("knocking about")
produced by the attractions of other stars, were invoked to
account for the development of the systems with longer periods
and elliptic orbits.
Sir James Jeans, in particular, has advocated this theory,
and readers are referred to his writings cited at the end of the
chapter for his mathematical development of it. But Jeans
himself has confirmed Liapounoff's conclusion* that the pear
shaped figure is unstable and has pointed out that if a double
star system results, it, too, will be unstable unless the mass ratio
of the two components is less than onethird. It has also been
shown by Moulton, Russell, and Jeans that even were a stable
double star system to result from fission, the mutual tidal actions
* Liapounoff, Mem. Imp. Acad. Set. St. Petersburg 17, 1905;
Jeans, Phil. Trans. Roy. Soc. A 217, 1, 1917.
278 THE BINARY STARS
of the two components could never greatly increase either the
major axis or the eccentricity of the orbit. Quite recently
William Markowitz* made a statistical investigation of the
possibility that shortperiod spectroscopic binaries might develop
into long period systems and in his work he had the benefit
of advice from Profs. MacMillan, Moulton, and Otto Struve,
He found that the necessary increase in the values of P, a, and e
could not result from contraction, secular decrease of mass, or
close encounters with other stars nor yet from the combination
of all three. Jeans had earlier concluded that pairs with periods
in excess of about 55 days could not result from the process of
fission, and that a different origin must be sought for them.
A separation of the binaries at the 55day period point or at
any other is negatived, in my judgment, by the fact that there
is no apparent correlation between period and mass,f and also
by the fact that there is a definite correlation between eccen
tricity and period. If the fission theory is incompetent to
explain the origin of the longer period binaries, that fact alone
leads us to abandon it entirely.
The separate nuclei theory remains to be considered. This
was first suggested by Laplace more than a hundred years ago.
In Note VII to his Systeme du Monde, he remarks :
"Such groups (as the Pleiades) are a necessary result of the
condensation of nebulae with several nuclei, for it is plain that
the nebulous matter being constantly attracted by these different
nuclei must finally form a group of stars like the Pleiades. The
condensation of nebulae with two nuclei will form stars in very
close proximity, which will turn one around the other similar to
* A p. Jour. 76, 69, 1932.
t In 1924, 1925, Jeans and E. W. Brown both found that, theoretically,
the semimajor axis (a) and therefore the period (P) would increase if one
or both components of a binary system were losing mass by radiation.
Jeans found that the eccentricity (e) would also remain constant, whereas
Brown's conclusion was that it, too, would increase though at a lower rate.
In a note in Nature for April 21, 1934, A. E. H. Bleksley states that a
recent investigation of his own confirms Jeans's results. He concludes
that the "semiaxis major is inversely proportional to the mass of the system"
throughout the life of a binary. He adds that a statistical study of all
available material shows that this relation appears to hold for the visual
binaries of known orbit but not for the shortperiod spectroscopic binaries,
and he suggests that there is a difference in origin between the two groups.
No details are given.
THE ORIGIN OF THE BINARY STARS 279
those double stars whose relative motions have already been
determined."*
The modern writer who has adopted this theory most explicitly
is the very man who first formulated the fission theory, Dr. See.
His discussion of the binary stars in the second volume of his
Researches on the Evolution of the Stellar System is devoted more
particularly to the development of the systems after their initial
formation but on page 232 we find the statement, "It is evident
. that the resulting massratio in a system depends on the
supply of nebulosity and the original nuclei already begun and
slowly developing in the nebula while it was still of vast extent
and great tenuity," and, on page 584, the even more definite
statement, "When a double star had been formed in the usual
manner by the growth of separate centers in a widely diffused
nebula. ..."
The separate nuclei theory apparently affords sufficient
latitude for the explanation of any binary system except, perhaps,
the very close, shortperiod spectroscopic binaries. To account
for these, the effect of a resisting medium has been invoked,
Markowitz, for example, finding that, unless the two components
are radiating mass more than 2.5 as fast as they are gathering
it in, the values of P, a, and e of the system are decreasing.
But even if we grant a tendency toward such a decrease, it is
hardly conceivable that it would account for the transformation
of, say, hundredyearperiod systems into systems with periods
of a single day, or of ten days, especially in view of the fact that
no correlation appears to exist between mass and period.
The fundamental objection to the separate nuclei theory is
that we really do not explain anything; to use Moulton's words,
"we only push by an assumption the problem of explaining the
binary systems a little farther back into the unknown."
Russell's specific objection, based upon his study of the
triple and quadruple systems, is also of great force. Why
should these systems almost invariably consist of a compara
tively close binary pair attended by a third star or by another
close binary pair at a distance relatively great? As Russell
says, "Not only is there no apparent reason for it, but if we
try to retrace in imagination the history of such a system,
through stages of greater and greater diffusion as we penetrate
*See Essays in Astronomy, p. 501 (edited by E. S. Holden, New York,
D. Appleton A'Company, 1900).
280 THE BINARY STARS
farther into the past (keeping in mind that the moment of
momentum of the whole system must remain constant), it is
hard to form any idea of the history of the nuclei which will
finally form a close and rapidly revolving pair, attended by a
distant companion."
The general conclusion of our discussion is that, although the
observational data that have been accumulated clearly indicate
the common origin of the binary stars, no theory of that origin
and of the subsequent development of the observed systems
that has so far been formulated can be regarded as satisfactory.
It is the duty of the observers to supply us with the data
required for the formulation and test of such a theory. Observa
tions of the various classes of binary systems, visual, spectro
scopic and eclipsing, must be continued indefinitely, by the
methods described in earlier chapters or by improved methods,
to provide the data needed for orbit computation. More orbits
of all these classes of systems are, of course, demanded. We
must have a number sufficiently great to insure confidence that
conclusions based upon our discussions are not affected by
selection in the data.
But that is not all. We are quite as much in need of accurate
values of the parallaxes of the systems and of accurate deter
minations of the magnitudes and spectral classes of the fainter
as well as of the brighter components. A knowledge of the
masses, the mass ratios, the absolute magnitudes, and the
spectral characteristics of the components in these systems is
quite as important as a knowledge of the orbital elements for the
solution of the problem of the origin and evolution of the binary
stars.
References
The following list contains only a few of the many papers that have been
read in preparing this chapter. References to others will be found in them,
and also in some of the papers cited at the end of Chap. VIII.
Darwin, G. H. : The Genesis of Double Stars, Darwin and Modern Science,
pp. 543564. Cambridge University Press, 1910.
: Presidential Address, British Association for the Advancement
of Science, Report B. A. A. S. 1905, p. 3.
Jeans, J. H. : The Motion of Tidally Distorted Masses with Special Refer
ence to Theories of Cosmogony. Mem. R.A.S. 62, part 1, 1917.
: On the Density of Algol Variables, Ap. Jour. 22, 93, 1905.
: The Evolution of Binary Systems, Mon. Not. R.A.S. 79, 100, 1918.
: The Origin of the Binary Stars, Scientia 31, 11, 1922.
THE ORIGIN OF THE BINARY STARS 281
: Astronomy and Cosmogony, pp. 198307, Cambridge University
Press, 1928. .
Moulton, F. R.: On Certain Relations among the Possible Changes in the
Motions of Mutually Attracting Spheres when Disturbed by Tidal
Interactions; and, Notes on the Possibility of fission of a Contracting
Rotating Fluid Mass, Publication 107, Carnegie Institution of Washing
ton, pp. 77160.
Introduction to Astronomy, revised edition, pp. 543548
Russell, H. N.: On the Origin of Binary Stars, Ap. Jour. 31, 185, 1910.
See, T. J. J.: Die Entwickelung der DoppelsternSysteme, Inaugural
Dissertation, 1892.
: Researches on the Physical Constitution of the Heavenly Bodies.
A.N. 169, 321, 1905.
: Researches on the Evolution of the Stellar Systems, 2, Chap. 20, 1910.
MacMillan, W. D.: The Problem of Two Bodies with Diminishing Mass,
Mon. Not. R.A.S. 86, 904, 1925.
Shajn, G.: On the MassRatio in Double Stars, Mon. Not. R.A.S. 86, 245,
1929; A.N. 287, 57, 1929.
Markowitz, W. : The Evolution of Binary Stars, Ap. Jour. 76, 69, 1932.
: Some Statistical and Dynamical Aspects of the Fission Theory,
Ap. Jour. 78, 161, 1933.
APPENDIX
The two tables printed on the following pages list the visual
and spectroscopic binary star orbits that have been used in the
preparation of the present volume. In compiling them, all
orbits available to me by September, 1933, have been examined.
The visual orbits listed in Table I have been taken from my
card catalogue of orbits, the orbits of the spectroscopic binaries,
from the card catalogue of variable radial velocities which
Dr. J. H. Moore keeps up to date.* I am deeply indebted
to Mrs. Moore for her kindness in copying off the necessary
data. Both sets of orbits have been checked by examination
of the original publications.
The abbreviations used for publications cited in the footnotes
and at the end of the chapters in the volume will, I think, be
readily understood. In the following tables, however, it has been
necessary to use the shortest possible form, and it may be well to
state that A.N., A.J., Ap. Jour., M.N., P.A., B.A.N., P.A.S.P.,
and R.A.S.C., stand, respectively, for AstronomischeNachrichten,
Astronomical Journal, Astrophysical Journal, Monthly Notices of
the Royal Astronomical Society, Popular Astronomy, Bulletin
of the Astronomical Institutes of the Netherlands, Publications
of the Astronomical Society of the Pacific, and Journal of the
Royal Astronomical Society of Canada. "Observatory," has been
abbreviated to 0., D.A.O., D.O. and A.O. standing, respectively,
for the Dominion Astrophysical, the Dominion and the Allegheny
Observatories. C.A. represents the Cape (of Good Hope)
Annals. The other abbreviations need no explanation.
* [Dr. Moore died on March 15, 1949. See p. 40.— J.T.K.]
283
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NAME INDEX
Adams, W. S., 228, 239, 242, 248
Airy, G., 17
Alter, D., 121, 244
Anderson, J. A., 68, 250
Andre, C, 123
Argelander, F. W. A., 264
Astrand, J. J., 80
Auwers, A., 118, 237, 238, 243
B
Baize, P., 40, 244
Baker, R. H., 182
Banachiewicz, T., 202
Barnard, E. E., 23, 47, 60, 240, 241
Barr, J. M., 213
Barton, S. G., 67
Bell, L., 270
Belopolsky, A., 134, 248, 254
Bergstrand, O., 244
Berman, L., 247, 248
Bernewitz, E., 97
Bessel, F. W., 13, 19, 118, 119n.,
225, 243
Bishop, G., 17
Bleksley, A. E. H., 278
Bode, J. E., 3, 4
Bohlin, K., 215
Bond, G. P., 1, 64, 237
Boothroyd, S. L., 22
Bos, W. H. van den, 37, 39, 60n.,
71, 78n., 79, 90, 112, 119, 204,
209n., 240, 246, 247, 260
Bosler, J., 62
Boss, L., 226, 227, 239
Bradley, J., 1, 2, 14, 243
Brashear, J. A., 37
Brill, A., 234
Brisbane, T. M., 35
Brown, E. W., 278n.
Burnham, S. W., 10, 17, 1922, 24,
25, 3235, 53, 59, 238, 240, 241
Campbell, W. W., 29, 46n., 135,
145, 150, 206, 210, 219, 249
Cannon, Annie J., 130, 267
Chamberlin, T. C., 275
Chandler, S. C, 182, 254
Chang, Y. C, 216
Clark, A., 17, 20, 23
Clark, A. G., 23, 118, 225, 237
Cogshall, W. A., 22
Comstock, G. C, 72, 111, 117, 120,
124, 228, 231
Cornu, M. A., 142, 149
Crawford, R. T., 114, 115
Crossley, E., 32
Curtis, H. D., 29, 159, 162, 165, 180,
248
Curtiss, R. H., 146, 150, 180, 254.
255
D
Danjon, M. A., 256n.
Darwin, G. H., 277, 280
Dawes, W. R., 17, 53, 57, 63, 72
Dawson, B. H., 37, 124, 244
De Sitter, W., 124
Dembowski, Ercole, 18, 19, 53, 72
Ditscheiner, L., 145
Doberck, W., 124, 203, 205, 206, 215,
235
Doig, P., 270n.
Donner, H. F., 38
Doolittle, Eric, 33, 34, 241
Doppler, Ch., 125
303
304
THE BINARY STARS
Dugan, R. S., 182, 198n.
Dunlop, J., 35
E
Eberhard, G., 66, 150
Eddington, A. S., 229, 233, 251, 252
Elkin, W. L., 47, 236
Encke, J. F., 71
Englemann, R., 19
Espin, T. E. H., 23, 34
Everett, Alice, 215
Fetlaar, J., 202
Feuille, L., 2
Finsen, W. S., 37, 102, 104, 105, 216,
235, 236, 240
Fizeau, H. L., 125
Flammarion, C, 32
Fletcher, A., 73n., 244
Fontenay, J. de, 1
Fox, Philip, 33, 64, 240
Fraunhofer, J., 11
Frost, E. B., 145, 150
Furner, H. F., 228
Harting, C. A. J., 183
Hartmann, J. F., 142, 147, 150, 251
Hartwig, E., 202
Heger, M. (Mrs. C. D. Shane), 252
Hellerich, J., 254, 256
Henroteau, F. C, 40, 114, 124, 175,
180
Henry, P., 64
Herschel, John, 9, 10, 15, 20, 25,
3236, 41, 71, 72, 76, 235, 243,
246
Herschel, William, 313, 15, 20, 25,
32, 33, 41, 70, 232, 246
Hertzsprung, E., 6466, 228, 231,
233, 246, 247, 271
Holden, E. S., 23, 33, 279».
Hooke, R., 1
Hough, G. W., 22, 32
Howard, C. P., 71, 123
Howe, H. A., 23
Huggins, William, 134
Humasson, M. L., 252
Humboldt, A. von, 237
Hussey, W. J., 17, 24, 25, 33, 37, 38,
57, 232, 245, 258
Huygens, C, 1
G
Galileo, G., 5
Gaposchkin, S., 201, 202
Gerasimovic, B. P., 252
Gill, David, 42, 236
Glasenapp, S., 71, 80, 89, 95, 97, 101
Gledhill, J., 19, 32
Goodericke, J., 27, 254
Gould, B. A., 64
Gyllenberg, W., 272
Innes, R. T. A., 3639, 54, 71, 79,
80, 90, 91, 94, 107, 111, 237, 240
Jacobi, K. G. J., 277
Jackson, J., 228, 231, 233, 272
Jeans, J., 277, 280
Jessup, M. K., 37
Jonckheere, R., 23, 34
Joy, A. H., 218n., 228, 248
Hagen, J. G., 202
Hall, Asaph, 31, 32, 123
Halley, E., 2
Halm, J. K. E., 180
Hargrave, L., 36
Hargreaves, F. J., 41n., 67
Harper, W. E., 177n.
K
Keeler, J. E., 150
Kepler, J., 93, 106, 113, 204
Kerrich, J. E., 124
King, E. S., 180
King, W. F., 157n., 169, 180
Kirchhoff, G. R., 129
NAME INDEX
305
Klinkerfues, E. F. W., 71, 123
Knott, G., 19
Kowalsky, M., 71, 80, 82, 89, 95, 97
Krat, W., 202
Kreiken, E. A., 215, 234, 272
Krueger, F., 241
Kuiper, G. P., 205n., 238, 242, 271
Kustner, F., 23, 33
Lacaille, N. L. de, 235
Lambert, J. H., 2
Lamont, R. P., 37
Laplace, P. S., 278
Lau, H. A., 270
Laves, K., 114, 175, 179
Leavenworth, F. P., 97
LehmannFilhes, R., 30, 158165,
179, 225
Leonard, F. C, 270
Leuschner, A. O., 70, 123
Lewis, T., 14, 15, 19, 33, 57, 58, 61,
121, 203, 215
Liapounoff, A. M., 277
Lohse, O., 235
Ludendorff, H., 206, 214, 233
Lundmark, K., 234, 272
LuplauJanssen, C, 204
Luyten, W. J., 248, 272
M
McClean, F., 36
McDiarmid, R. J., 196
McLaughlin, D. B., 255, 256
Maclaurin, C., 277
MacMillan, W. D., 275278, 281
Maderni, A., 180
Madler, J. H., 19, 71
Maggini, M., 68, 69, 97
Markowitz, W., 278, 279, 281
Maury, Antonia C, 28, 130
Maxwell, A. D., 245ro.
Mayer, C., 3, 4
Mayer, T., 232
Merrill, P. W., 68, 250, 252
Meyermann, B., 124
Michell, J., 2, 8
Michelson, A. A., 68, 250
Milburn, W., 23
Montanari, G., 253
Moore, Charlotte E., 230, 231
Moore, J. H., 29, 125, 155n., 180, 239
Moulton, F. R., 114, 275, 278, 281
Miiller, G., 202
N
Nassau, J. J., 124
Newall, H. F., 150, 249
Newton, Isaac, 70
Nijland, A. N., 179, 182
Norlund, N. E., 119, 246
O
Olivier, C. P., 67
Oort, J. H., 271
Orloff, A., 180
Paddock, G. F., 177n., 179, 180
Parkhurst, J. A., 182
Parvulesco, C, 124
Pearce, J. A., 252, 253
Peters, C. A. F., 118, 237
Peters, C. F. W., 244
Piazzi, G., 243
Picart, L., 180
Pickering, E. C, 1, 26, 64, 130, 182,
183, 201, 254, 258».
Pitman, J. H., 234
Plaskett, J. S., 150, 252
Plummer, H. t 179, 180
Pogo, A., 180, 214
Poincare, H., 277
Pollock, J. A., 36
Poor, J. M., 216
Pound, J., 1, 2, 245
Prager, R., 202
Przybyllok, E., 246
Ptolemy, 2
R
Rambaut, A. A., 30, 123, 164, 179
Redman, R. O., 231
306
THE BINARY STARS
Reese, H. M., 250
Riccioli, J. B., 1
Richaud, J., 1
Roberts, A. W., 182, 183, 201
Rossiter, R. A., 37, 255
Rowland, H. A., 141, 146
Russell, H. C, 36
Russell, H. N., 71, 80, 86, 117, 118,
124, 164, 179, 183, 184, 188,
192, 197, 201, 208, 226n., 228
231, 277, 279, 281
S
Safford, T. H., 237
Salet, P., 62
Sanford, R. S., 248
Savary, F., 71
Schaeberle, J. M., 118
Scharbe, 202
Scheiner, J., 67, 134
Schiaparelli, G. V., 19, 31, 59
Schlesinger, F., 63, 121, 155, 175n.,
176, 180, 206, 214, 223, 236,
244, 245n., 254, 255
Schnauder, M., 216
Schorr, R. H., 123
Schuster, A., 56
Schwarzschild, K., 68, 71, 123, 156,
165, 168, 179
Seares, F. H., 182, 242
Secchi, A., 19, 130
See, T. J. J., 22, 71, 108, 205, 215,
235, 277, 279, 281
Seeliger, H. von, 71, 119, 123, 260
262
Sellors, R. P., 36
Shajn, G., 215, 233, 234, 270»., 281
Shapley, H., 183, 197, 201, 202
Sitterly, B. W., 183, 202
Slipher, V. M., 252
Smythe, W. H., 18
South, J., 9, 10, 243w.
Stebbins, Joel, 182, 231, 255
Stein, J., 67, 202
Stone, O., 23
Stoney, G. J., 275
Struve, F. G. W., 1120, 25, 33, 36,
72, 246, 249, 270
Struve, H:, 62
Struve, O., 17, 19, 24, 25, 33, 61,
114, 245
Struve, Otto, 204, 205, 214, 252
Thiele, T. N., 71, 79, 8992, 94, 101,
107, 111, 123
Tisserand, F. F., 183
Turner, H. H., 215
Vahlen, T., 124
Van Biesbroeck, G., 33, 97, 107
Van Gent, H., 248
Villarceau, Yvon, 71
Vogel, H. C., 27, 28, 134, 150, 254
Vogt, H., 202
Volet, C., 80n., 124, 238, 240, 241
Voute, J., 62
Vyssotsky, A., 238, 240
W
Wallenquist, A., 271
Walter, K., 202
Webb, T. W., 21, 22
Wendell, O. C., 182, 192, 238
Wilkins, P. O., 124
Wilsing, J., 164, 179
Wilson, J. M., 19, 32
Wright, W. H., 119, 136, 138, 150,
225, 236, 247, 256
Young, R. K., 154n., 155n., 173n.
Z
Zagar, F., 240
Zurhellen, W., 156, 165, 179
Zwiers, H. J., 71, 80, 86, 95, 100, 101
SUBJECT INDEX
Algol, 253
Angstrom unit defined, 129
B
Binary stars, definition of, ix
density of, 231
distribution of, by spectral class,
267269
invisible companions in systems
of, 118, 225
masses of, 216
multiple systems of, 224, 232, 245,
274
number of known orbits of, 203
205
parallaxes of, from orbits and
radial velocities, 224
relations between period, and
eccentricity in orbits of, 205,
213
and spectral class in orbits of,
210
systems of special interest, 235
tables, of the known orbits of,
283, 288
of masses of, 220
theories of the origin of, capture
theory, 275
fission theory, 277
general statement of problem
of, 273
separate nuclei theory, 278
summary of, 280
(See also Double stars; Eclipsing
binary stars; Spectroscopic
binary stars; Visual binary
stars)
Cancri, Tau, 232
CapeUa, 249
Castor, 248
Centauri, Alpha, 235
Cepheid variable stars, 30
Color contrast in double stars, 16,
270
61 Gygni, 243
D
DopplerFizeau principle, 126
Double stars, Burnham's discoveries
and measures, 20, 25
color contrast in, 16, 270
conventions for measures of, 15, 41
correction for proper motion in, 73
diaphragms used in measuring, 59
distribution of, by angular dis
tances, and magnitudes, etc.,
261
in right ascension and declina
tion, 260
early discoveries of, 1
early orbit methods for, 71, 72
early speculations on character of,
2
errors of measure of, 60
eyepieces for measuring, 59
first collection of, 3
first photograph of, 1, 64
general catalogues of, 32, 39
Herschel's discoveries and theo
ries, 4
Herschel (J) and South's work on,
9
interferometer measures of, 67
Lick Observatory survey for dis
covery of, 24, 257
307
308
THE BINARY STARS
Double stars, magnitude estimates
of components of, 54
methods of measuring, 49
observing program for, 55
percentage of, among stars near
the Sun, 267
by magnitude classes, 265
photographic measures of, 64
southern hemisphere work on, 35
Struve (F. G. W.) and the
Mensurae Micrometricae, 12
Struve (O.) and the Pulkowa
Catalogue, 17
working definition of, 257
(See also Binary stars; Visual
binary stars)
Driving clock, the first, 11, 12
Dynamical parallaxes, 227
E
Eclipsing binary stars, density of,
198
early history of, 181
example of orbit of, 192
reflection effect in, 198
Russell's method of computing the
orbits of, 183
table of orbits of, 200
Equulei, Delta, 244
Eyepieces, methods of finding the
magnifying power of, 59
G
Gravitation, universality of the law
of, 70
H
Hercvlis, Zeta, 120
Interferometer measures of double
stars, 67
K
Kirchhoff's law, 129
Krueger 60, 241
lick Observatory double star survey,
24, 257
M
Masses, 216
table of, 220
Mass function, 217223
Micrometer, the comparison image,
41
the filar, 41
Micrometer screw, value of revolu
tion of, 44
Multiple systems, 120, 224, 232, 245,
274
O
70 Ophiuchi, 120
Orbit of A 88, 95
of 24 Aqr., 101
of 42 C. Br., 113
of W Del., 192
of € Equ., 117
of k Vel., 162
Parallel, method of determining, 43
Photographic methods of measuring
double stars, 64
R
Radial velocity of a star, chapter on
the, by J. H. Moore, 125
early determinations of the, 134
tests for correctness of measures
of, 149
(See also Spectrograms; Spectro
graphs; Spectroscopes)
Rectilinear motion, 121
Resolving power, of a spectroscope,
128
of a telescope, 56
S
Sirius, 237
SUBJECT INDEX
309
Spectrograms, CornuHartmann for
mulae for, 142
dispersion curves for reducing, 142
example of the measurement and
reduction of, 141
spectrocomparator method of re
duction of, 147
velocitystandard method of re
duction of, 146
Spectrograph, description of the
Mills, 135
loss of light in a stellar, 139
Spectroscope, essential parts of, 127
resolving power of, 128
Spectroscopic binary stars, cata
logues of, 29
differential correction of the ele
ments of, 175
discoveries of, 1, 26, 39
distribution of, by spectral class,
269
co in orbits of, 213
equation for the radial velocity of,
152
example of orbit determination of,
162
first solution of the orbit problem,
30
H and K lines in, 251
masses of the, 217
methods of orbit computation,
conditions of the problem,
151
fundamental formulae, 153, 157
King's method, 169
LehmannFilhes's method, 158
Russell's short method, 174
Schwarzschild's method, 165
Zurhellen's methods, 166
orbits of, with small eccentricity,
178
secondary oscillations in orbits of,
177
Spectroscopic binary stars, table
of orbits of, 288
velocity curve for, 154, 157
(See also Binary stars; Eclipsing
binary stars)
Stellar radial velocities, determina
tion of, 134
Stellar spectra, classification of, 130
U
Ursae Majoris, Xi, 119, 246248
Variable proper motion, 118, 237
Visual binary stars, apparent ellipse,
from elements of, 108
from observations of, 72
definition of orbit elements of, 77
differential corrections to elements
of, 109
distribution of longitudes of peri,
helion in, 213
dynamical parallaxes of, 227
formulae, for ephemerides for, 79,
80
for orbits when i = 90°, 113
mass ratios in, 226
methods for computing orbits of,
GlasenappKowalsky method,
80,95
special methods, 113
ThieleInnes constants and
methods, 89, 101
Zwiers's method, 86, 100
orientation of the orbit planes of,
215
statistical data relating to the, 257
table of orbits of the, 284
(See also Double stars)
Catalogue of Dover
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of thermodynamics, other problems posed by discarding of Laplacean determinism. 285pp.
5% x 8. T480 Paperbound SI .65
THE PRINCIPLES OF SCIENCE, A TREATISE ON LOGIC AND THE SCIENTIFIC METHOD, W. S.
Jevons. Milestone in development of symbolic logic remains stimulating contribution to in
vestigation of inferential validity in sciences. Treats inductive, deductive logic, theory of
number, probability, limits of scientific method; significantly advances Boole's logic, con
tains detailed introduction to nature and methods of probability in physics, astronomy,
everyday affairs, etc. In introduction, Ernest Nagel of Columbia U. says, "[Jevons] continues
to be of interest as an attempt to articulate the logic of scientific inquiry." liii + 786pp.
5% x 8. S446 Paperbound $2.98
A HISTORY OF ASTRONOMY FROM THALES TO KEPLER, J. L. E. Dreyer. Only work in English
to give complete history of cosmological views from prehistoric times to Kepler. Partial
contents: Near Eastern astronomical systems, Early Greeks, Homocentric spheres of
Euxodus, Epicycles, Ptolemaic system, Medieval cosmology, Copernicus, Kepler, much more.
"Especially useful to teachers and students of the history of science . . . unsurpassed in
its field," Isis. Formerly "A History of Planetary Systems from Thales to Kepler." Revised
foreword by W. H. Stahl. xvii + 430pp. 5% x 8. S79 Paperbound $1.98
A CONCISE HISTORY OF MATHEMATICS, D. Struik. Lucid study of development of ideas,
techniques, from Ancient Near East, Greece, Islamic science, Middle Ages, Renaissance,
modern times. Important mathematicians described in detail. Treatment not anecdotal, but
analytical development of ideas. Nontechnical — no math training needed. "Rich in con
tent, thoughtful in interpretations," U.S. Quarterly Booklist. 60 illustrations including
Greek, Egyptian manuscripts, portraits of 31 mathematicians. 2nd edition, xix + 299pp.
5% x 8. S255 Paperbound $1.75
THE PHILOSOPHICAL WRITINGS OF PEIRCE, edited by Justus Buchler. A carefully balanced
expositon of Peirce's complete system, written by Peirce himself. It covers such matters
as scientific method, pure chance vs. law, symbolic logic, theory of signs, pragmatism,
experiment, and other topics. "Excellent selection . . . gives more than adequate evidence
of the range and greatness," Personalis! Formerly entitled "The Philosophy of Peirce."
xvi + 368pp. T217 Paperbound $1.95
SCIENCE AND METHOD, Henri Poincart. Procedure of scientific discovery, methodology, ex
periment, ideagermination — processes by which discoveries come into being. Most signifi
cant and interesting aspects of development, application of ideas. Chapters cover selection
of facts, chance, mathematical reasoning, mathematics and logic; Whitehead, Russell,
Cantor, the new mechanics, etc. 288pp. 5% x 8. S222 Paperbound $1.35
SCIENCE AND HYPOTHESIS, Henri Poincart. Creative psychology in science. How such con
cepts as number, magnitude, space, force, classical mechanics develooed, how modern
scientist uses them in his thought. Hypothesis in physics, theories of modern physics.
Introduction by Sir James Larmor. "Few mathematicians have had the breadth of vision
of Poincar6, and none is his superior in the gift of clear exposition," E. T. Bell. 272pp.
5% x 8. S221 Paperbound $1.35
ESSAYS IN . EXPERIMENTAL LOGIC, John Dewey. Stimulating series of essays by one of most
influential minds in American philosophy presents some of his most mature thoughts on
wide range of subjects. Partial contents: Relationship between inquiry and experience;
dependence of knowledge upon thought; character logic; judgments of practice, data, and
meanings; stimuli of thought, etc. viii + 444pp. 5% x 8. T73 Paperbound $1.95
WHAT IS SCIENCE, Norman Campbell. Excellent introduction explains scientific method, role
of mathematics, types of scientific laws. Contents: 2 aspects of science, science and
nature, laws of chance, discovery of laws, explanation of laws, measurement and numerical
laws, applications of science. 192pp. 5% x 8. S43 Paperbound $1.25
CATALOGUE OF
FROM EUCLID TO EDDINGTON: A STUDY OF THE CONCEPTIONS OF THE EXTERNAL WORLD, Sir
Edmund Whittaker. Foremost British scientist traces development of theories of natural phi
losophy from western rediscovery of Euclid to Eddington, Einstein, Dirac, etc. 5 major
divisions: Space, Time and Movement; Concepts of Classical Physics; Concepts of Quantum
Mechanics; Eddington Universe. Contrasts inadequacy of classical physics to understand
physical world with present day attempts of relativity, nonEuclidean geometry, space
curvature, etc. 212pp. 5% x 8. T491 Paperbound $1.35
THE ANALYSIS OF MATTER, Bertrand Russell. How do our senses accord with the new
physics? This volume covers such topics as logical analysis of physics, prerelativity
physics, causality, scientific inference, physics and perception, special and general rela
tivity, Weyl's theory, tensors, invariants and their physical interpretation, periodicity and
qualitative series. "The most thorough treatment of the subject that has yet been pub
lished," The Nation. Introduction by L. E. Denonn. 422pp. 5% x 8. T231 Paperbound $1.95
LANGUAGE, TRUTH, AND LOGIC, A. Ayer. A clear introduction to the Vienna and Cambridge
schools of Logical Positivism. Specific tests to evaluate validity of ideas, etc. Contents:
function of philosophy, elimination of metaphysics, nature of analysis, a priori, truth and
probability, etc. 10th printing. "I should like to have written it myself," Bertrand Russell.
160pp. 5% x 8. T10 Paperbound $1.25
THE PSYCHOLOGY OF INVENTION IN THE MATHEMATICAL FIELD, J. Hadamard. Where do ideas
come from? What role does the unconscious play? Are ideas best developed by mathematical
reasoning, word reasoning, visualization? What are the methods used by Einstein, Poincare,
Galton, Riemann? How can these techniques be applied by others? One of the world's
leading mathematicians discusses these and other questions, xiii + 145pp. 5% x 8.
T107 Paperbound $1.25
GUIDE TO PHILOSOPHY, C. E. M. Joad. By one of the ablest expositors of all time, this is
not simply a history or a typological survey, but an examination of central problems in
terms of answers afforded by the greatest thinkers: Plato, Aristotle, Scholastics, Leibniz,
Kant, Whitehead, Russell, and many others. Especially valuable to persons in the physical
sciences; over 100 pages devoted to Jeans, Eddington, and others, the philosophy of
modern physics, scientific materialism, pragmatism, etc. Classified bibliography. 592pp.
5% x 8. T50 Paperbound $2.00
SUBSTANCE AND FUNCTION, and EINSTEIN'S THEORY OF RELATIVITY, Ernst Cassirer. Two
books bound as one. Cassirer establishes a philosophy of the exact sciences that takes into
consideration new developments in mathematics, shows historical connections. Partial
contents: Aristotelian logic, Mill's analysis, Helmholtz and Kronecker, Russell and cardinal
numbers, Euclidean vs. nonEuclidean geometry, Einstein's relativity. Bibliography. Index,
xxi + 464pp. 5% x 8. T50 Paperbound $2.00
FOUNDATIONS OF GEOMETRY, Bertrand Russell. Nobel laureate analyzes basic problems in
the overlap area between mathematics and philosophy: the nature of geometrical knowledge,
the nature of geometry, and the applications of geometry to space. Covers history of non
Euclidean geometry, philosophic interpretations of geometry, especially Kant, projective
and metrical geometry. Most interesting as the solution offered in 1897 by a great mind
to a problem still current. New introduction by Prof. Morris Kline, N.Y. University. "Ad
mirably clear, precise, and elegantly reasoned analysis," International Math. News, xii +
201pp. 5% x 8. S233 Paperbound $1.60
THE NATURE OF PHYSICAL THEORY, P. W. Bridgman. How modern physics looks to a highly
unorthodox physicist — a Nobel laureate. Pointing out many absurdities of science, demon
strating inadequacies of various physical theories, weighs and analyzes contributions of
Einstein, Bohr, Heisenberg, many others. A nontechnical consideration of correlation of
science and reality, xi I 138pp. 5% x 8. S33 Paperbound $1.25
EXPERIMENT AND THEORY IN PHYSICS, Max Born. A Nobel laureate examines the nature
and value of the counterclaims of experiment and theory in physics. Synthetic versus
analytical scientific advances are analyzed in works of Einstein, Bohr, Heisenberg, Planck,
Eddington, Milne, others, by a fellow scientist. 44pp. 5% x 8. S308 Paperbound 60$
A SHORT HISTORY OF ANATOMY AND PHYSIOLOGY FROM THE GREEKS TO HARVEY, Charles
Singer. Corrected edition of "The Evolution of Anatomy." Classic traces anatomy, phys
iology from prescientific times through Greek, Roman periods, dark ages, Renaissance, to
beginning of modern concepts. Centers on individuals, movements, that definitely advanced
anatomical knowledge. Plato, Diodes, Erasistratus, Galen, da Vinci, etc. Special section
on Vesalius. 20 plates. 270 extremely interesting illustrations of ancient, Medieval, Renais
sance, Oriental origin, xii + 209pp. 5% x 8. T389 Paperbound $1.75
SPACE TIME MATTER, Hermann Weyl. "The standard treatise on the general theory of
relativity," (Nature), by world renowned scientist. Deep, clear discussion of logical coher
ence of general theory, introducing all needed tools: Maxwell, analytical geometry, non
Euclidean geometry, tensor calculus, etc. Basis is classical spacetime, before absorption
of relativity. Contents: Euclidean space, mathematical form, metrical continuum, general
theory, etc. 15 diagrams, xviii + 330pp. 5% x 8. S267 Paperbound $1.75
DOVER SCIENCE BOOKS
MATTER AND MOTION, James Clerk Maxwell. Excellent exposition begins with simple par
ticles, proceeds gradually to physical systems beyond complete analysis; motion, force
properties of centre of mass of material system; work, energy, gravitation, etc. Written
with all Maxwell's original insights and clarity. Notes by E. Larmor. 17 diagrams. 178pp.
5% x 8  S188 Paperbound $1.25
PRINCIPLES OF MECHANICS, Heinrich Hertz. Last work by the great 19th century physicist
is not only a classic, but of great interest in the logic of science. Creating a new system
of mechanics based upon space, time, and mass, it returns to axiomatic analysis, under
standing of the formal or structural aspects of science, taking into account logic, observa
tion, a priori elements. Of great historical importance to Poincare, Carnap, Einstein, Milne
A 20 page introduction by R. S. Cohen, Wesleyan University, analyzes the implications of
gerte s thought and the logic of science. 13 page introduction by Helmholtz. xlii + 274pp
5% x 8  S316 Clothbound $3.50
S317 Paperbound $1.75
FROM MAGIC TO SCIENCE, Charles Singer. A great historian examines aspects of science
from Roman Empire through Renaissance. Includes perhaps best discussion of early herbals
penetrating physiological interpretation of "The Visions of Hildegarde of Bingen." Also
examines Arabian, Galenic influences; Pythagoras' sphere, Paracelsus; reawakening of
science under Leonardo da Vinci, Ves^alius; Lorica of Gildas the Briton; etc. Frequent
quotations with translations from contemporary manuscripts. Unabridged, corrected edi
tion. 158 unusual illustrations from Classical, Medieval sources, xxvii + 365pp. 5% x 8.
T390 Paperbound $2.00
A HISTORY OF THE CALCULUS, AND ITS CONCEPTUAL DEVELOPMENT, Carl B. Boyer. Provides
laymen, mathematicians a detailed history of the development of the calculus, from begin
nings in antiquity to final elaboration as mathematical abstraction. Gives a sense of
mathematics not as technique, but as habit of mind, in progression of ideas of Zeno, Plato
Pythagoras, Eudoxus, Arabic and Scholastic mathematicians, Newton, Leibniz, Taylor, Des
cartes, Euler, Lagrange, Cantor, Weierstrass, and others. This first comprehensive, critical
history of the calculus was originally entitled "The Concepts of the Calculus." Foreword
by R. Courant. 22 figures. 25 page bibliography, v + 364pp. 5% x 8.
S509 Paperbound $2.00
A DIDEROT PICTORIAL ENCYCLOPEDIA OF TRADES AND INDUSTRY, Manufacturing and the
Technical Arts in Plates Selected from "L'Encyclopfidie ou Dictionnaire Raisonni des
Sciences, des Arts, et des Metiers" of Denis Diderot. Edited with text by C. Gillispie First
modern selection of plates from highpoint of 18th century French engraving. Storehouse
of technological information to historian of arts and science. Over 2,000 illustrations on
485 full page plates, most of them original size, show trades, industries of fascinating
era in such great detail that modern reconstructions might be made of them. Plates teem
with men, women, children performing thousands of operations; show sequence general
operations, closeups, details of machinery. Illustrates such important, interesting trades
industries as sowing, harvesting, beekeeping, tobacco processing, fishing, arts of war'
mining, smelting, casting iron, extracting mercury, making gunpowder, cannons, bells'
shoeing horses, tanning, papermaking, printing, dying, over 45 more categories. Professor
Gillispie of Princeton supplies full commentary on all plates, identifies operations, tools
processes, etc. Material is presented in lively, lucid fashion. Of great interest to all
studying history of science, technology. Heavy library cloth. 920pp. 9 x 12.
T421 2 volume set $18.50
DE MAGNETE, William Gilbert. Classic work on magnetism, founded new science. Gilbert
was first to use word "electricity," to recognize mass as distinct from weight, to discover
effect of heat on magnetic bodies; invented an electroscope, differentiated between static
electricity and magnetism, conceived of earth as magnet. This lively work, by first great
experimental scientist, is not only a valuable historical landmark, but a delightfully easy
to follow record of a searching, ingenious mind. Translated by P. F. Mottelay. 25 page
biographical memoir. 90 figures, lix + 368pp. 5% x 8. S470 Paperbound $2.00
HISTORY OF MATHEMATICS, D. E. Smith. Most comprehensive, nontechnical history of math
in English. Discusses lives and works of over a thousand major, minor figures, with foot
notes giving technical information outside book's scheme, and indicating disputed matters
Vol. I: A chronological examination, from primitive concepts through Egypt Babylonia
Greece, the Orient, Rome, the Middle Ages, The Renaissance, and to 1900. Vol. II The
development of ideas in specific fields and problems, up through elementary calculus.
"Marks an epoch . . . will modify the entire teaching of the history of science," George
Sarton. 2 volumes, total of 510 illustrations, 1355pp. 5% x 8. Set boxed in attractive
container. T429, 430 Paperbound, the set $5.00
THE PHILOSOPHY OF SPACE AND TIME, H. Reichenbach. An important landmark in develop
ment of empiricist conception of geometry, covering foundations of geometry, time theory,
consequences of Einstein's relativity, including: relations between theory and observations
coordinate definitions; relations between topological and metrical properties of space;
psychological problem of visual intuition of nonEuclidean structures; many more topics
important to modern science and philosophy. Majority of ideas require only knowledge of
intermediate math. "Still the best book in the field," Rudolf Carnap. Introduction by
R. Carnap. 49 figures, xviii + 296pp. 5% x 8. S443 Paperbound $2.00
CATALOGUE OF
FOUNDATIONS OF SCIENCE: THE PHILOSOPHY OF THEORY AND EXPERIMENT, N. Campbell.
A critique of the most fundamental concepts of science, particularly physics. Examines why
certain propositions are accepted without question, demarcates science from philosophy,
etc. Part I analyzes presuppositions of scientific thought: existence of material world,
nature of laws, probability, etc; part 2 covers nature of experiment and applications of
mathematics: conditions for measurement, relations between numerical laws and theories,
error, etc. An appendix covers problems arising from relativity, force, motion, space,
time. A classic in its field. "A real grasp of what science is," Higher Educational Journal,
xiii + 565pp. 5% x 83/fe. S372 Paperbound $2.95
THE STUDY OF THE HISTORY OF MATHEMATICS and THE STUDY OF THE HISTORY OF SCIENCE,
G. Sarton. Excellent introductions, orientation, for beginning or mature worker. Describes
duty of mathematical historian, incessant efforts and genius of previous generations. Ex
plains how today's discipline differs from previous methods. 200 item bibliography with
critical evaluations, best available biographies of modern mathematicians, best treatises
on historical methods is especially valuable. 10 illustrations. 2 volumes bound as one.
113pp. + 75pp. 5% x 8. T240 Paperbound $1.25
MATHEMATICAL PUZZLES
MATHEMATICAL PUZZLES OF SAM LOYD, selected and edited by Martin Gardner. 117 choice
puzzles by greatest American puzzle creator and innovator, from his famous "Cyclopedia
of Puzzles." All unique style, historical flavor of originals. Based on arithmetic, algebra,
probability, game theory, route tracing, topology, sliding block, operations research, geo
metrical dissection. Includes famous "1415" puzzle which was national craze, "Horse of
a Different Color" which sold millions of copies. 120 line drawings, diagrams. Solutions.
xx + 167pp. 5% x 8. T498 Paperbound $1.00
SYMBOLIC LOGIC and THE GAME OF LOGIC, Lewis Carroll. "Symbolic Logic" is not concerned
with modern symbolic logic, but is instead a collection of over 380 problems posed with
charm and imagination, using the syllogism, and a fascinating diagrammatic method of
drawing conclusions. In "The Game of Logic" Carroll's whimsical imagination devises a
logical game played with 2 diagrams and counters (included) to manipulate hundreds of
tricky syllogisms. The final section, "Hit or Miss" is a lagniappe of 101 additional puzzles
in the delightful Carroll manner. Until this reprint edition, both of these books were rarities
costing up to $15 each. Symbolic Logic: Index, xxxi + 199pp. The Game of Logic: 96pp.
2 vols, bound as one. 5% x 8. T492 Paperbound $1.50
PILLOW PROBLEMS and A TANGLED TALE, Lewis Carroll. One of the rarest of all Carroll's
works, "Pillow Problems" contains 72 original math puzzles, all typically ingenious. Particu
larly fascinating are Carroll's answers which remain exactly as he thought them out,
reflecting his actual mental process. The problems in "A Tangled Tale" are in story form,
originally appearing as a monthly magazine serial. Carroll not only gives the solutions, but
uses answers sent in by readers to discuss wrong approaches and misleading paths, and
grades them for insight. Both of these books were rarities until this edition, "Pillow
Problems" costing up to $25, and "A Tangled Tale" $15. Pillow Problems: Preface and
Introduction by Lewis Carroll, xx + 109pp. A Tangled Tale: 6 illustrations. 152pp. Two vols,
bound as one. 5% x 8. T493 Paperbound $1.50
NEW WORD PUZZLES, G. L. Kaufman. 100 brand new challenging puzzles on words, com
binations, never before published. Most are new types invented by author, for beginners
and experts both. Squares of letters follow chess moves to build words; symmetrical
designs made of synonyms; rhymed crostics; double word squares; syllable puzzles where
you fill in missing syllables instead of missing letter; many other types, all new. Solutions.
"Excellent," Recreation. 100 puzzles. 196 figures, vi + 122pp. 5% x 8.
T344 Paperbound $1.00
MATHEMATICAL EXCURSIONS, H. A. Merrill. Fun, recreation, insights into elementary prob
lem solving. Math expert guides you on bypaths not generally travelled in elementary math
courses — divide by inspection, Russian peasant multiplication; memory systems for pi; odd,
even magic squares; dyadic systems; square roots by geometry; Tchebichev's machine;
dozens more. Solutions to more difficult ones. "Brain stirring stuff ... a classic," Genie.
50 illustrations. 145pp. 5% x 8. T350 Paperbound $1.00
THE BOOK OF MODERN PUZZLES, G. L. Kaufman. Over 150 puzzles, absolutely all new mate
rial based on same appeal as crosswords, deduction puzzles, but with different principles,
techniques. 2minute teasers, word labyrinths, design, pattern, logic, observation puzzles,
puzzles testing ability to apply general knowledge to peculiar situations, many others.
Solutions. 116 illustrations. 192pp. 5% x 8. T143 Paperbound $1.00
MATHEMAGIC, MAGIC PUZZLES, AND GAMES WITH NUMBERS, R. V. Heath. Over 60 puzzles,
stunts, on properties of numbers. Easy techniques for multiplying large numbers mentally,
identifying unknown numbers, finding date of any day in any year. Includes The Lost Digit,
3 Acrobats, Psychic Bridge, magic squares, trjangles, cubes, others not easily found else
where. Edited by J. S. Meyer. 76 illustrations. 128pp. 5% x 8. T110 Paperbound $1.00
DOVER SCIENCE BOOKS
PUZZLE QUIZ AND STUNT FUN, J. Meyer. 238 highpriority puzzles, stunts, tricks— math
puzzles like The Clever Carpenter, Atom Bomb, Please Help Alice; mysteries, deductions
like The Bridge of Sighs, Secret Code; observation puzzlers like The American Flag, Playing
Cards, Telephone Dial; over 200 others with magic squares, tongue twisters, puns, ana
grams. Solutions. Revised, enlarged edition of "FunToDo." Over 100 illustrations. 238
puzzles, stunts, tricks. 256pp. 5% x 8. T337 Paperbound $1.00
101 PUZZLES IN THOUGHT AND LOGIC, C. R. Wylie, Jr. For readers who enjoy challenge,
stimulation of logical puzzles without specialized math or scientific knowledge. Problems
entirely new, range from relatively easy to brainteasers for hours of subtle entertainment.
Detective puzzles, find the lying fisherman, how a blind man identifies color by logic, many
more. Easytounderstand introduction to logic of puzzle solving and general scientific
method. 128pp. 5% x 8. T367 Paperbound $1.00
CRYPTANALYSIS, H. F. Gaines. Standard elementary, intermediate text for serious students.
Not just old material, but much not generally known, except to experts. Concealment,
Transposition, Substitution ciphers; Vigenere, Kasiski, Playfair, multafid, dozens of other
techniques. Formerly "Elementary Cryptanalysis." Appendix with sequence charts, letter
frequencies in English, 5 other languages, English word frequencies. Bibliography. 167
codes. New to this edition: solutions to codes, vi + 230pp. 5% x 8%.
T97 Paperbound $1.95
CRYPTOGRAPY, L. D. Smith. Excellent elementary introduction to enciphering, deciphering
secret writing. Explains transposition, substitution ciphers; codes; solutions; geometrical
patterns, route transcription, columnar transposition, other methods. Mixed cipher systems;
single, polyalphabetical substitutions; mechanical devices; Vigenere; etc. Enciphering Jap
anese; explanation of Baconian biliteral cipher; frequency tables. Over 150 problems. Bib
liography. Index. 164pp. 5% x 8. T247 Paperbound $1.00
MATHEMATICS, MAGIC AND MYSTERY, M. Gardner. Card tricks, metal mathematics, stage
mindreading, other "magic" explained as applications of probability, sets, number theory,
etc. Creative examination of laws, applications. Scores of new tricks, insights. 115 sections
on cards, dice, coins; vanishing tricks, many others. No sleight of hand — math guarantees
success. "Could hardly get more entertainment . . . easy to follow," Mathematics Teacher.
115 illustrations, xii + 174pp. 5% x 8. T335 Paperbound $1.00
AMUSEMENTS IN MATHEMATICS, H. E. Dudeney. Foremost British originator of math puzzles,
always witty, intriguing, paradoxical in this classic. One of largest collections. More than
430 puzzles, problems, paradoxes. Mazes, games, problems on number manipulations,
unicursal, other route problems, puzzles on measuring, weighing, packing, age, kinship,
chessboards, joiners', crossing river, plane figure dissection, many others. Solutions. More
than 450 illustrations, viii + 258pp. 5% x 8. T473 Paperbound $1.25
THE CANTERBURY PUZZLES H. E. Dudeney. Chaucer's pilgrims set one another problems in
story form. Also Adventures of the Puzzle Club, the Strange Escape of the King's Jester,
the Monks of Riddlewell, the Squire's Christmas Puzzle Party, others. All puzzles are
original, based on dissecting plane figures, arithmetic, algebra, elementary calculus, other
branches of mathematics, and purely logical ingenuity. "The limit of ingenuity and in
tricacy," The Observer. Over 110 puzzles, full solutions. 150 illustrations, viii + 225 pp.
5% x 8. T474 Paperbound $1.25
MATHEMATICAL PUZZLES FOR BEGINNERS AND ENTHUSIASTS, G. MottSmith. 188 puzzles to
test mental agility. Inference, interpretation, algebra, dissection of plane figures, geometry,
properties of numbers, decimation, permutations, probability, all are in these delightful
problems. Includes the Odic Force, How to Draw an Ellipse, Spider's Cousin, more than 180
others. Detailed solutions. Appendix with square roots, triangular numbers, primes, etc.
135 illustrations. 2nd revised edition. 248pp. 5% x 8. T198 Paperbound $1.00
MATHEMATICAL RECREATIONS, M. Kraitchik. Some 250 puzzles, problems, demonstrations of
recreation mathematics on relatively advanced level. Unusual historical problems from
Greek, Medieval, Arabic, Hindu sources; modem problems on "mathematics without num
bers," geometry, topology, arithmetic, etc. Pastimes derived from figurative, Mersenne,
Fermat numbers: fairy chess; latruncles: reversi; etc. Full solutions. Excellent insights
into special fields of math. "Strongly recommended to all who are interested in the
lighter side of mathematics," Mathematical Gaz. 181 illustrations. 330pp. 5% x 8.
T163 Paperbound $1.75
FICTION
FLATLAND, E. A. Abbott. A perennially popular sciencefiction classic about life in a 2
dimensional world, and the impingement of higher dimensions. Political, satiric, humorous,
moral overtones. This land where women are straight lines and the lowest and most dan
gerous classes are isosceles triangles with 3° vertices conveys brilliantly a feeling for
many concepts of modern science. 7th edition. New introduction by Banesh Hoffmann. 128pp.
5% x 8 Tl Paperbound $1.00
CATALOGUE OF
SEVEN SCIENCE FICTION NOVELS OF H. 6. WELLS. Complete texts, unabridged, of seven of
Wells' greatest novels: The War of the Worlds, The Invisible Man, The Island of Or. Moreau,
The Food of the Gods, First Men in the Moon, In the Days of the Comet, The Time Machine.
Still considered by many experts to be the best sciencefiction ever written, they will offer
amusements and instruction to the scientific minded reader. "The great master," Sky and
Telescope. 1051pp. 5% x 8. T264 Clothbound $3.95
28 SCIENCE FICTION STORIES OF H. 6. WELLS. Unabridged! This enormous omnibus contains
2 full length novels — Men Like Gods, Star Begotten — plus 26 short stories of space, time,
invention, biology, etc. The Crystal Egg, The Country of the Blind, Empire of the Ants,
The Man Who Could Work Miracles, Aepyornis Island, A Story of the Days to Come, and
20 others "A master ... not surpassed by . . . writers of today," The English Journal.
915pp. 5% x 8. T265 Clothbound $3.95
FIVE ADVENTURE NOVELS OF H. RIDER HAGGARD. All the mystery and adventure of darkest
Africa captured accurately by a man who lived among Zulus for years, who knew African
ethnology, folkways as did few of his contemporaries. They have been regarded as examples
of the very best high adventure by such critics as Orwell, Andrew Lang, Kipling. Contents:
She, King Solomon's Mines, Allan Quatermain, Allan's Wife, Maiwa's Revenge. "Could spin
a yarn so full of suspense and color that you couldn't put the story down," Sat. Review.
821pp. 5% x 8. T108 Clothbound $3.95
CHESS AND CHECKERS
LEARN CHESS FROM THE MASTERS, Fred Reinfeld. Easiest, most instructive way to im
prove your game — play 10 games against such masters as Marshall, ZnoskoBorovsky, Bron
stein, Najdorf, etc., with each move graded by easy system. Includes ratings for alternate
moves possible. Games selected for interest, clarity, easily isolated principles. Covers
Ruy Lopez, Dutch Defense, Vienna Game openings; subtle, intricate middle game variations;
allimportant end game. Full annotations. Formerly "Chess by Yourself." 91 diagrams, viii
+ 144pp. 5% x 8. T362 Paperbound $1.00
REINFELD ON THE END GAME IN CHESS, Fred Reinfeld. Analyzes 62 end games by Alekhine,
Flohr, Tarrasch, Morphy, Capablanca, Rubinstein, Lasker, Reshevsky, other masters. Only
1st rate book with extensive coverage of error— tell exactly what is wrong with each move
you might have made. Centers around transitions from middle play to end play. King and
pawn, minor pieces, queen endings; blockage, weak, passed pawns, etc. "Excellent ... a
boon," Chess Life. Formerly "Practical End Play." 62 figures, vi + 177pp. 5% x 8.
T417 Paperbound $1.25
HYPERMODERN CHESS as developed in the games of its greatest exponent, ARON NIMZO
VICH, edited by Fred Reinfeld. An intensely original player, analyst, Nimzovich's approaches
startled, often angered the chess world. This volume, designed for the average player,
shows how his iconoclastic methods won him victories over Alekhine, Lasker, Marshall,
Rubinstein, Spielmann, others, and infused new life into the game. Use his methods to
startle opponents, invigorate play. "Annotations and introductions to each game ... are
excellent," Times (London). 180 diagrams, viii + 220pp. 5% x 8. T448 Paperbound $1.35
THE ADVENTURE OF CHESS, Edward Lasker. Lively reader, by one of America's finest chess
masters, including: history of chess, from ancient Indian 4handed game of Chaturanga
to great players of today; such delights and oddities as Maelzel's chessplaying automaton
that beat Napoleon 3 times; etc. One of most valuable features is author's personal recollec
tions of men he has played against — Nimzovich, Emanuel Lasker, Capablanca, Alekhine,
etc. Discussion of chessplaying machines (newly revised). 5 page chess primer. 11 illus
trations. 53 diagrams. 296pp. 5% x 8. S510 Paperbound $1.45
THE ART OF CHESS, James Mason. Unabridged reprinting of latest revised edition of most
famous general study ever written. Mason, early 20th century master, teaches beginning,
intermediate player over 90 openings; middle game, end game, to see more moves ahead,
to plan purposefully, attack, sacrifice, defend, exchange, govern general strategy. "Classic
... one of the clearest and best developed studies," Publishers Weekly. Also included, a
complete supplement by F. Reinfeld, "How Do You Play Chess?", invaluable to beginners
for its lively questionandanswer method. 448 diagrams. 1947 ReinfeldBernstein text.
Bibliography, xvi + 340pp. 5% x 8. T463 Paperbound $1.85
MORPHY'S GAMES OF CHESS, edited by P. W. Sergeant. Put boldness into your game by
flowing brilliant, forceful moves of the greatest chess player of all time. 300 of Morphy's
best games, carefully annotated to reveal principles. 54 classics against masters like
Anderssen, Harrwitz, Bird, Paulsen, and others. 52 games at odds; 54 blindfold games; plus
over 100 others. Follow his interpretation of Dutch Defense, Evans Gambit, Giuoco Piano,
Ruy Lopez, many more. Unabridged reissue of latest revised edition. New introduction by
F. Reinfeld. Annotations, introduction by Sergeant. 235 diagrams, x + 352pp. 5% x 8.
T386 Paperbound $1.75
8
DOVER SCIENCE BOOKS
WIN AT CHECKERS, M. Hopper. (Formerly "Checkers.") Former World's Unrestricted Checker
Champion discusses principles of game, expert's shots, traps, problems for beginner, stand
ard openings, locating best move, end game, opening "blitzkrieg" moves to draw when
behind, etc. Over 100 detailed questions, answers anticipate problems. Appendix. 75 prob
lems with solutions, diagrams. 79 figures, xi + 107pp. 5% x 8. T363 Paperbound $1.00
HOW TO FORCE CHECKMATE, Fred Reinfeld. If you have trouble finishing off your opponent,
here is a collection of lightning strokes and combinations from actual tournament play.
Starts with 1move checkmates, works up to 3move mates. Develops ability to lock ahead,
gain new insights into combinations, complex or deceptive positions, ways to estimate weak
nesses, strengths of you and your opponent. "A good deal of amusement and instruction,"
Times, (London). 300 diagrams. Solutions to all positions. Formerly "Challenge to Chess
Players." 111pp. 5% x 8. T417 Paperbound 1.25
A TREASURY OF CHESS LORE, edited by Fred Reinfeld. Delightful collection of anecdotes,
short stories, aphorisms by, about masters, poems, accounts of games, tournaments, photo
graphs; hundreds of humorous, pithy, satirical, wise, historical episodes, comments, word
portraits. Fascinating "must" for chess players; revealing and perhaps seductive to those
who wonder what their friends see in game. 49 photographs (14 full page plates). 12
diagrams, xi + 306pp. 5% x 8. T458 Paperbound SI. 75
WIN AT CHESS, Fred Reinfeld. 300 practical chess situations, to sharpen your eye, test skill
against masters. Start with simple examples, progress at own pace to complexities. This
selected series of crucial moments in chess will stimulate imagination, develop stronger,
more versatile game. Simple grading system enables you to judge progress. "Extensive use
of diagrams is a great attraction," Chess. 300 diagrams. Notes, solutions to every situation.
Formerly "Chess Quiz." vi + 120pp. 5% x 8. T433 Paperbound $1.00
MATHEMATICS:
ELEMENTARY TO INTERMEDIATE
HOW TO CALCULATE QUICKLY, H. Sticker. Tried and true method to help mathematics of
everyday life. Awakens "number sense"— ability to see relationships between numbers as
whole quantities. A serious course of over 9000 problems and their solutions through
techniques not taught in schools: lefttoright multiplications, new fast division, etc. 10
minutes a day will double or triple calculation speed. Excellent for scientist at home in
higher math, but dissatisfied with speed and accuracy in lower math. 256pp. 5 x 7V*.
Paperbound $1.00
FAMOUS PROBLEMS OF ELEMENTARY GEOMETRY, Felix Klein. Expanded version of 1894
Easter lectures at Gottingen. 3 problems of classical geometry: squaring the circle, trisect
ing angle, doubling cube, considered with full modern implications: transcendental num
bers, pi, etc. "A modern classic ... no knowledge of higher mathematics is required,"
Scientia. Notes by R. Archibald. 16 figures, xi + 92pp. 5% x 8. T298 Paperbound $1.00
HIGHER MATHEMATICS FOR STUDENTS OF CHEMISTRY ANO PHYSICS, J. W. Mellor. Practical,
not abstract, building problems out of familiar laboratory material. Covers differential cal
culus, coordinate, analytical geometry, functions, integral calculus, infinite series, numerical
equations, differential equations, Fourier's theorem probability, theory of errors, calculus
of variations, determinants. "If the reader is not familiar with this book, it will repay
him to examine it," Chem. and Engineering News. 800 problems. 189 figures, xxi + 641pp.
5% x 8. S193 Paperbound $2.25
TRIGONOMETRY REFRESHER FOR TECHNICAL MEN, A. A. Klaf. 913 detailed questions, answers
cover most important aspects of plane, spherical trigonometry — particularly useful in clearing
up difficulties in special areas. Part I: plane trig, angles, quadrants, functions, graphical repre
sentation, interpolation, equations, logs, solution of triangle, use of slide rule, etc. Next
188 pages discuss applications to navigation, surveying, elasticity, architecture, other
special fields. Part 3: spherical trig, applications to terrestrial, astronomical problems.
Methods of timesaving, simplification of principal angles, make book most useful. 913
questions answered. 1738 problems, answers to odd numbers. 494 figures. 24 pages of for
mulas, functions, x + 629pp. 5% x 8. T371 Paperbound $2.00
CALCULUS REFRESHER FOR TECHNICAL MEN, A. A. Klaf. 756 questions examine most im
portant aspects of integral, differential calculus. Part I: simple differential calculus, con
stants, variables, functions, increments, logs, curves, etc. Part 2: fundamental ideas of
integrations, inspection, substitution, areas, volumes, mean value, double, triple integration,
etc. Practical aspects stressed. 50 pages illustrate applications to specific problems of civil,
nautical engineering, electricity, stress, strain, elasticity, similar fields. 756 questions
answered. 566 problems, mostly answered. 36pp. of useful constants, formulas, v + 431pp.
5% x 8. T370 Paperbound $2.00
CATALOGUE OF
MONOGRAPHS ON TOPICS OF MODERN MATHEMATICS, edited by J. W. A. Young. Advanced
mathematics for persons who have forgotten, or not gone beyond, high school algebra
9 monographs on foundation of geometry, modern pure geometry, nonEuclidean geometry,
fundamental propositions of algebra, algebraic equations, functions, calculus, theory of
numbers, etc. Each monograph gives proofs of important results, and descriptions of lead
ing methods, to provide wide coverage. "Of high merit," Scientific American. New intro
duction by Prof. M. Kline, N.Y. Univ. 100 diagrams, xvi + 416pp. 6Vs x 9V4.
S289 Paperbound $2.00
MATHEMATICS IN ACTION, 0. G. Sutton. Excellent middle level application of mathematics
to study of universe, demonstrates how math is applied to ballistics, theory of computing
machines, waves, wavelike phenomena, theory of fluid flow, meteorological problems,
statistics, flight, similar phenomena. No knowledge of advanced math required. Differential
equations, Fourier series, group concepts, Eigentunctions, Planck's constant, airfoil theory,
and similar topics explained so clearly in everyday language that almost anyone can derive
benefit from reading this even if much of highschool math is forgotten. 2nd edition. 88
figures, viii + 236pp. 5% x 8. T450 Clothbound $3.50
ELEMENTARY MATHEMATICS FROM AN ADVANCED STANDPOINT, Felix Klein. Classic text,
an outgrowth of Klein's famous integration and survey course at Gottingen. Using one field
to interpret, adjust another, it covers basic topics in each area, with extensive analysis.
Especially valuable in areas of modern mathematics. "A great mathematician, inspiring
teacher, . . . deep insight," Bui., Amer. Math Soc.
Vol. I. ARITHMETIC, ALGEBRA, ANALYSIS. Introduces concept of function immediately, en
livens discussion with graphical, geometric methods. Partial contents: natural numbers,
special properties, complex numbers. Real equations with real unknowns, complex quan
tities. Logarithmic, exponential functions, infinitesimal calculus. Transcendence of e and pi,
theory of assemblages. Index. 125 figures, ix + 274pp. 5% x 8. S151 Paperbound $1.75
Vol. II. GEOMETRY. Comprehensive view, accompanies space perception inherent in geom
etry with analytic formulas which facilitate precise formulation. Partial contents: Simplest
geometric manifold; line segments, Grassman determinant principles, classication of con
figurations of space. Geometric transformations: affine, projective, higher point transforma
tions, theory of the imaginary. Systematic discussion of geometry and its foundations. 141
illustrations, ix + 214pp. 5% x 8. S151 Paperbound $1.75
A TREATISE ON PLANE AND ADVANCED TRIGONOMETRY, E. W. Hobson. Extraordinarily wide
coverage, going beyond usual college level, one of few works covering advanced trig in
full detail. By a great expositor with unerring anticipation of potentially difficult points.
Includes circular functions; expansion of functions of multiple angle; trig tables; relations
between sides, angles of triangles; complex numbers; etc. Many problems fully solved.
"The best work on the subject," Nature. Formerly entitled "A Treatise on Plane Trigonom
etry." 689 examples. 66 figures, xvi + 383pp. 5% x 8. S353 Paperbound $1.95
NONEUCLIDEAN GEOMETRY, Roberto Bonola. The standard coverage of nonEuclidean geom
etry. Examines from both a historical and mathematical point of view geometries which
have arisen from a study of Euclid's 5th postulate on parallel lines. Also included are
complete texts, translated, of Bolyai's "Theory of Absolute Space," Lobachevsky's "Theory
of Parallels." 180 diagrams. 431pp. 5% x 8. S27 Paperbound $1.95
GEOMETRY OF FOUR DIMENSIONS, H. P. Manning. Unique in English as a clear, concise intro
duction. Treatment is synthetic, mostly Euclidean, though in hyperplanes and hyperspheres
at infinity, nonEuclidean geometry is used. Historical introduction. Foundations of 4dimen
sional geometry. Perpendicularity, simple angles. Angles of planes, higher order. Symmetry,
order, motion; hyperpyramids, hypercones, hyperspheres; figures with parallel elements;
volume, hypervolume in space; regular polyhedroids. Glossary. 78 figures, ix + 348pp.
5 % x 8  S182 Paperbound $1.95
MATHEMATICS: INTERMEDIATE TO ADVANCED
GEOMETRY (EUCLIDEAN AND NONEUCLIDEAN)
THE GEOMETRY OF RENE' DESCARTES. With this book, Descartes founded analytical geometry.
Original French text, with Descartes's own diagrams, and excellent SmithLatham transla
tion. Contains: Problems the Construction of Which Requires only Straight Lines and Circles;
On the Nature of Curved Lines; On the Construction of Solid or Supersolid Problems. Dia
grams. 258pp. 5% x 8. S68 Paperbound $1.50
10
DOVER SCIENCE BOOKS
THE WORKS OF ARCHIMEDES, edited by T. L. Heath. Ail the known works of the great Greek
mathematician, including the recently discovered Method of Archimedes. Contains: On
Sphere and Cylinder, Measurement of a Circle, Spirals, Conoids, Spheroids, etc. Definitive
edition of greatest mathematical intellect of ancient world. 186 page study by Heath dis
cusses Archimedes and history of Greek mathematics. 563pp. 5% x 8. S9 Paperbound $2.00
COLLECTED WORKS OF BERNARD RIEMANN. Important sourcebook, first to contain complete
text of 1892 "Werke" and the 1902 supplement, unabridged. 31 monographs, 3 complete
lecture courses, 15 miscellaneous papers which have been of enormous importance in
relativity, topology, theory of complex variables, other areas of mathematics. Edited by
R. Dedekind, H. Weber, M. Noether, W. Wirtinger. German text; English introduction by
Hans Lewy. 690pp. 5% x 8. S226 Paperbound $2.85
THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS, edited by Sir Thomas Heath. Definitive edition
of one of very greatest classics of Western world. Complete translation of Heiberg text,
plus spurious Book XIV. 150 page introduction on Greek, Medieval mathematics, Euclid,
texts, commentators, etc. Elaborate critical apparatus parallels text, analyzing each defini
tion, postulate, proposition, covering textual matters, refutations, supports, extrapolations,
etc. This is the full Euclid. Unabridged reproduction of Cambridge U. 2nd edition. 3 vol
umes. 995 figures. 1426pp. 5% x 8. S88, 89, 90, 3 volume set, paperbound $6.00
AN INTRODUCTION TO GEOMETRY OF N DIMENSIONS, D. M. Y. Sommerville. Presupposes no
previous knowledge of field. Only book in English devoted exclusively to higher dimensional
geometry. Discusses fundamental ideas of incidence, parallelism, perpendicularity, angles
between linear space, enumerative geometry, analytical geometry from projective and metric
views, polytopes, elementary ideas in analysis situs, content of hyperspacial figures. 60
diagrams. 196pp. 5% x 8. S494 Paperbound $1.50
ELEMENTS OF NONEUCLIDEAN GEOMETRY, D. M. Y. Sommerville. Unique in proceeding step
bystep. Requires only good knowledge of highschool geometry and algebra, to grasp ele
mentary hyperbolic, elliptic, analytic nonEuclidean Geometries; space curvature and its
implications; radical axes; homopethic centres and systems of circles; parataxy and parallel
ism; Gauss' proof of defect area theorem; much more, with exceptional clarity. 126 prob
lems at chapter ends. 133 figures, xvi + 274pp. 5% x 8. S460 Paperbound $1.50
THE FOUNDATIONS OF EUCLIDEAN GEOMETRY, H. G. Forder. First connected, rigorous ac
count in light of modern analysis, establishing propositions without recourse to empiricism,
without multiplying hypotheses. Based on tools of 19th and 20th century mathematicians,
who made it possible to remedy gaps and complexities, recognize problems not earlier
discerned. Begins with important relationship of number systems in geometrical figures.
Considers classes, relations, linear order, natural numbers, axioms for magnitudes, groups,
quasiflelds, fields, nonArchimedian systems, the axiom system (at length), particular axioms
(two chapters on the Parallel Axioms), constructions, congruence, similarity, etc. Lists.
axioms employed, constructions, symbols in frequent use. 295pp. 53/% x 8.
S481 Paperbound $2.00
CALCULUS, FUNCTION THEORY (REAL AND COMPLEX),
FOURIER THEORY
FIVE VOLUME "THEORY OF FUNCTIONS" SET BY KONRAD KNOPP. Provides complete, readily
followed account of theory of functions. Proofs given concisely, yet without sacrifice of
completeness or rigor. These volumes used as texts by such universities as M.I.T., Chicago,
N.Y. City College, many others. "Excellent introduction . . . remarkably readable, concise,
clear, rigorous," J. of the American Statistical Association.
ELEMENTS OF THE THEORY OF FUNCTIONS, Konrad Knopp. Provides background for further
volumes in this set, or texts on similar level. Partial contents: Foundations, system of com
plex numbers and Gaussian plane of numbers, Riemann sphere of numbers, mapping by
linear functions, normal forms, the logarithm, cyclometric functions, binomial series. "Not
only for the young student, but also for the student who knows all about what is in it,"
Mathematical Journal. 140pp. 5% x 8. S154 Paperbound $1.35
THEORY OF FUNCTIONS, PART I, Konrad Knopp. With volume II, provides coverage of basic
concepts and theorems. Partial contents: numbers and points, functions of a complex
variable, integral of a continuous function, Cauchy's intergral theorem, Cauchy's integral
formulae, series with variable terms, expansion and analytic function in a power series,
analytic continuation and complete definition of analytic '"ictions, Laurent expansion, types
of singularities, vii + 146pp. 5% x 8. S156 Paperbound $1.35
THEORY OF FUNCTIONS, PART II, Konrad Knopp. Application and further development of
general theory, special topics. Single valued functions, entire, Weierstrass. Meromorphic
functions: MittagLeffler. Periodic functions. Multiple valued functions. Riemann surfaces.
Algebraic functions. Analytical configurations, Riemann surface, x + 150pp. 5% x 8.
S157 Paperbound $1.35
11
CATALOGUE OF
PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME I, Konrad Knopp. Problems in ele
mentary theory, for use with Knopp's "Theory of Functions," or any other text. Arranged
according to increasing difficulty. Fundamental concepts, sequences of numbers and infinite
series, complex variable, integral theorems, development in series, conformal mapping.
Answers, viii + 126pp. 5% x 8. S 158 Paperbound $1.35
PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME II, Konrad Knopp. Advanced theory
of functions, to be used with Knopp's "Theory of Functions," or comparable text. Singular
ities, entire and meromorphic functions, periodic, analytic, continuation, multiplevalued
functions, Riemann surfaces, conformal mapping. Includes section of elementary problems
"The difficult task of selecting . . . problems just within the reach of the beginner is
here masterfully accomplished," AM. MATH. SOC. Answers. 138pp. 5*/b x 8.
S159 Paperbound $1.35
ADVANCED CALCULUS, E. B. Wilson. Still recognized as one of most comprehensive, useful
texts. Immense amount of wellrepresented, fundamental material, including chapters on
vector functions, ordinary differential equations, special functions, calculus of variations
etc., which are excellent introductions to these areas. Requires only one year of calculus'
Over 1300 exercises cover both pure math and applications to engineering and physical
problems. Ideal reference, refresher. 54 page introductory review, ix + 566pp. 5% x 8
S504 Paperbound $2.45
LECTURES ON THE THEORY OF ELLIPTIC FUNCTIONS, H. Hancock. Reissue of only book in
English with so extensive a coverage, especially of Abel, Jacobi, Legendre, Weierstrass
Hermite, Liouville, and Riemann. Unusual fullness of treatment, plus applications as well as
theory in discussing universe of elliptic integrals, originating in works of Abel and
Jacobi. Use is made of Riemann to provide most general theory. 40page table of formulas
76 figures, xxiii + 498pp. 5% x 8. S483 Paperbound $2.55
THEORY OF FUNCTIONALS AND OF INTEGRAL AND INTEGRODIFFERENTIAL EQUATIONS, Vito
Volterra. Unabridged republication of only English translation, General theory of functions
depending on continuous set of values of another function. Based on author's concept of
transition from finite number of variables to a continually infinite number. Includes much
material on calculus of variations. Begins with fundamentals, examines generalization of
analytic functions, functional derivative equations, applications, other directions of theory
etc. New introduction by G. C. Evans. Biography, criticism of Volterra's work by E Whit
taker, xxxx + 226pp. 5% x 8. S502 Paperbound $1.75
AN INTRODUCTION TO FOURIER METHODS AND THE LAPLACE TRANSFORMATION, Philip
Franklin. Concentrates on essentials, gives broad view, suitable for most applications. Re
quires only knowledge of calculus. Covers complex qualities with methods of computing ele
mentary functions for complex values of argument and finding approximations by charts;
Fourier series; harmonic anaylsis; much more. Methods are related to physical problems
of heat flow, vibrations, electrical transmission, electromagnetic radiation, etc. 828 prob
lems, answers. Formerly entitled "Fourier Methods." x + 289pp. 5% x 8.
S452 Paperbound $1.75
THE ANALYTICAL THEORY OF HEAT, Joseph Fourier. This book, which revolutionized mathe
matical physics, has been used by generations of mathematicians and physicists interested
in heat or application of Fourier integral. Covers cause and reflection of rays of heat,
radiant heating, heating of closed spaces, use of trigonometric series in theory of heat,
Fourier integral, etc. Translated by Alexander Freeman. 20 figures, xxii + 466pp. 5% x 8.
S93 Paperbound $2.00
ELLIPTIC INTEGRALS, H. Hancock. Invaluable in work involving differential equations with
cubics, quatrics under root sign, where elementary calculus methods are inadequate. Prac
tical solutions to problems in mathematics, engineering, physics; differential equations re
quiring integration of Lam6's, Briot's, or Bouquet's equations; determination of arc of
ellipse, hyperbola, lemiscate; solutions of problems in elastics; motion of a projectile under
resistance varying as the cube of the velocity; pendulums; more. Exposition in accordance
with LegendreJacobi theory. Rigorous discussion of Legendre transformations. 20 figures.
5 place table. 104pp. 5% x 8. S484 Paperbound $1.25
THE TAYLOR SERIES, AN INTRODUCTION TO THE THEORY OF FUNCTIONS OF A COMPLEX
VARIABLE, P. Dienes. Uses Taylor series to approach theory of functions, using ordinary
calculus only, except in last 2 chapters. Starts with introduction to real variable and com
plex algebra, derives properties of infinite series, complex differentiation, integration, etc.
Covers biuniform mapping, overconvergence and gap theorems, Taylor series on its circle
of convergence, etc. Unabridged corrected reissue of first edition. 186 examples, many
fully worked out. 67 figures, xii + 555pp. 5% x 8. S391 Paperbound $2.75
LINEAR INTEGRAL EQUATIONS, W. V. Lovitt. Systematic survey of general theory, with some
application to differential equations, calculus of variations, problems of math, physics.
Includes: integral equation of 2nd kind by successive substitutions; Fredholm's equation
as ratio of 2 integral series in lambda, applications of the Fredholm theory, HilbertSchmidt
theory of symmetric kernels, application, etc. Neumann, Dirichlet, vibratory problems,
ix + 253pp. 5% x 8. S175 Clothbound $3.50
S176 Paperbound $1.60
12
DOVER SCIENCE BOOKS
DICTIONARY OF CONFORMAL REPRESENTATIONS, H. Kober. Developed by British Admiralty to
solve Laplace's equation in 2 dimensions. Scores of geometrical forms and transformations
for electrical engineers, Joukowski aerofoil for aerodynamics, SchwartzChristoffel trans
formations for hydrodynamics, transcendental functions. Contents classified according to
analytical functions describing transformations with corresponding regions. Glossary. Topo
logical index. 447 diagrams. 6Va x 9V*. S160 Paperbound $2.00
ELEMENTS OF THE THEORY OF REAL FUNCTIONS, J. E. Littlewood. Based on lectures at
Trinity College, Cambridge, this book has proved extremely successful in introducing graduate
students to modern theory of functions. Offers full and concise coverage of classes and
cardinal numbers, well ordered series, other types of series, and elements of the theory
of sets of points. 3rd revised edition, vii + 71pp. 5% x 8. S171 Clothbound $2.85
S172 Paperbound $1.25
INFINITE SEQUENCES AND SERIES, Konrad Knopp. 1st publication in any language. Excellent
introduction to 2 topics of modern mathematics, designed to give student background to
penetrate further alone. Sequences and sets, real and complex numbers, etc. Functions of
a real and complex variable. Sequences and series. Infinite series. Convergent power series.
Expansion of elementary functions. Numerical evaluation of series, v + 186pp. 5% x 8.
5152 Clothbound 53.50
5153 Paperbound $1.75
THE THEORY AND FUNCTIONS OF A REAL VARIABLE AND THE THEORY OF FOURIER'S SERIES,
E. W .Hobson. One of the best introductions to set theory and various aspects of functions
and Fourier's series. Requires only a good background in calculus. Exhaustive .coverage of:
metric and descriptive properties of sets of points; transfinite numbers and order types;
functions of a real variable; the Riemann and Lebesgue integrals; sequences and series
of numbers; powerseries; functions representable by series sequences of continuous func
tions; trigonometrical series; representation of functions by Fourier's series; and much
more. "The best possible guide," Nature. Vol. I: 88 detailed examples, 10 figures. Index,
xv + 736pp. Vol. II: 117 detailed examples, 13 figures, x + 780pp. 6% * t 9V4. _, MM
Vol. I: S387 Paperbound $3.00
Vol. II: S388 Paperbound $3.00
ALMOST PERIODIC FUNCTIONS, A. S. Besicovitch. Unique and important summary by a well
known mathematician covers in detail the two stages of development in Bohr's theory
of almost periodic functions: (1) as a generalization of pure periodicity, with results and
proofs; (2) the work done by Stepanof, Wiener, Weyl, and Bohr in generalizing the theory,
xi + 180pp. 5% x 8. S18 Paperbound $1.75
INTRODUCTION TO THE THEORY OF FOURIER'S SERIES AND INTEGRALS, H. S. Carslaw. 3rd
revised edition, an outgrowth of author's courses at Cambridge. Historical introduction,
rational, irrational numbers, infinite sequences and series, functions of a single variable,
definite integral, Fourier series, and similar topics. Appendices discuss practical harmonic
analysis, periodogram analysis, Lebesgue's theory. 84 examples, xiii + 368pp. 5% x»,
S48 Paperbound $2.00
SYMBOLIC LOGIC
THE ELEMENTS OF MATHEMATICAL LOGIC, Paul Rosenbloom. First publication in any lan
guage. For mathematically mature readers with no training in symbolic, logic. Development
of lectures given at Lund Univ., Sweden, 1948. Partial contents: Logic of classes, funda
mental theorems, Boolean algebra, logic of propositions, of propositional functions, expres
sive languages, combinatory logics, development of math within an object language, para
doxes, theorems of Post, Goedel, Church, and similar topics, iv + 214pp. 5% x 8.
S227 Paperbound $1.45
INTRODUCTION TO SYMBOLIC LOGIC AND ITS APPLICATION, R. Carnap. Clear, comprehensive,
rigorous, by perhaps greatest living master. Symbolic languages analyzed, one constructed.
Applications to math (axiom systems for set theory, real, natural numbers), topology
(Dedekind, Cantor continuity explanations), physics (general analysis of determination, cau
sality, spacetime topology), biology (axiom system for basic concepts). "A masterpiece,
Zentralblatt fur Mathematik und Ihre Grenzgebiete. Over 300 exercises. 5 figures, xvi +
241pp. 5% x 8. S453 Paperbound $1.85
AN INTRODUCTION TO SYMBOLIC LOGIC, Susanne K. Langer. Probably clearest book for the
philosopher, scientist, layman — no special knowledge of math required. Starts with simplest
symbols, goes on to give remarkable grasp of BooleSchroeder, RussellWhitehead systems,
clearly, quickly. Partial Contents: Forms, Generalization, Classes, Deductive System of
Classes, Algebra of Logic, Assumptions of Principia Mathematica, Logistics, Proofs of
Theorems, etc. "Clearest . . . simplest introduction .'. . the intelligent nonmathematician
should have no difficulty," MATHEMATICS GAZETTE. Revised, expanded 2nd edition. Truth
value tables. 368pp. 5% 8. S164 Paperbound $1.75
13
CATALOGUE OF
TRIGONOMETRICAL SERIES, Anton i Zygmund. On modern advanced level. Contains carefully
organized analyses of trigonometric, orthogonal, Fourier systems of functions, with clear
adequate descriptions of summability of Fourier series, proximation theory, conjugate series,
convergence, divergence of Fourier series. Especially valuable for Russian, Eastern Euro
pean coverage. 329pp. 5% x 8. S290 Paperbound $1.50
THE LAWS OF THOUGHT, George Boole. This book founded symbolic logic some 100 years
ago. It is the 1st significant attempt to apply logic to all aspects of human endeavour.
Partial contents: derivation of laws, signs and laws, interpretations, eliminations, condi
tions of a perfect method, analysis, Aristotelian logic, probability, and similar topics,
xvii + 424pp. 5% x 8. S28 Paperbound $2.00
SYMBOLIC LOGIC, C. I. Lewis, C. H. Langford. 2nd revised edition of probably most cited
book in symbolic logic. Wide coverage of entire field; one of fullest treatments of paradoxes;
plus much material not available elsewhere. Basic to volume is distinction between logic
of extensions and intensions. Considerable emphasis on converse substitution, while matrix
system presents supposition of variety of nonAristotelian logics. Especially valuable sec
tions on strict limitations, existence theorems. Partial contents: BooleSchroeder algebra
truth value systems, the matrix method; implication and deductibility; general theory of
propositions; etc. "Most valuable," Times, London. 506pp. 5% x 8. S170 Paperbound $2.00
GROUP THEORY AND LINEAR ALGEBRA, SETS, ETC.
LECTURES ON THE ICOSAHEDRON AND THE SOLUTION OF EQUATIONS OF THE FIFTH DEGREE,
Felix Klein. Solution of quintics in terms of rotations of regular icosahedron around its
axes of symmetry. A classic, indispensable source for those interested in higher algebra,
geometry, crystallography. Considerable explanatory material included. 230 footnotes, mostly
bibliography. "Classical monograph . . . detailed, readable book," Math. Gazette. 2nd edi
tion, xvi + 289pp. 5% x 8. S314 Paperbound $1.85
INTRODUCTION TO THE THEORY OF GROUPS OF FINITE ORDER, R. Carmichael. Examines
fundamental theorems and their applications. Beginning with sets, systems, permutations,
etc., progresses in easy stages through important types of groups: Abelian, prime power,
permutation, etc. Except 1 chapter where matrices are desirable, no higher math is needed.
783 exercises, problems, xvi + 447pp. 5% x 8. S299 Clothbound $3.95
S300 Paperbound $2.00
THEORY OF GROUPS OF FINITE ORDER, W. Burnside. First published some 40 years ago,
still one of clearest introductions. Partial contents: permutations, groups independent of
representation, composition series of a group, isomorphism of a group with itself, Abelian
groups, prime power groups, permutation groups, invariants of groups of linear substitu
tion, graphical representation, etc. "Clear and detailed discussion . . . numerous problems
which are instructive," Design News, xxiv + 512pp. 5% x 8. S38 Paperbound $2.45
COMPUTATIONAL METHODS OF LINEAR ALGEBRA, V. N. Faddeeva, translated by C. D. Benster.
1st English translation of unique, valuable work, only one in English presenting systematic
exposition of most important methods of linear algebra — classical, contemporary. Details
of deriving numerical solutions of problems in mathematical physics. Theory and practice.
Includes survey of necessary background, most important methods of solution, 'for exact,
iterative groups. One of most valuable features is 23 tables, triple checked for accuracy,
unavailable elsewhere. Translator's note, x + 252pp. 5% x 8. S424 Paperbound $1.95
THE CONTINUUM AND OTHER TYPES OF SERIAL ORDER, E. V. Huntington. This famous book
gives a systematic elementary account of the modern theory of the continuum as a type
of serial order. Based on the CantorDedekind ordinal theory, which requires no technical
knowledge of higher mathematics, it offers an easily followed analysis of ordered classes,
discrete and dense series, continuous series, Cantor's trahsfinite numbers. "Admirable
introduction to the rigorous theory of the continuum . . . reading easy," Science Progress.
2nd edition, viii + 82pp. 5% x 8. S129 Clothbound $2.75
S130 Paperbound $1.00
THEORY OF SETS, E. Kamke. Clearest, amplest introduction in English, well suited for inde
pendent study. Subdivisions of main theory, such as theory of sets of points, are discussed,
but emphasis is on general theory. Partial contents: rudiments of set theory, arbitrary sets,
their cardinal numbers, ordered sets, their order types, wellordered sets, their cardinal
numbers, vii 4 144pp. 5% x 8. S141 Paperbound $1.35
CONTRIBUTIONS TO THE FOUNDING OF THE THEORY OF TRANSFINITE NUMBERS, Georg Cantor.
These papers founded a new branch of mathematics. The famous articles of 18957 are
translated, with an 82page introduction by P. E. B. Jourdain dealing with Cantor, the
background of his discoveries, their results, future possibilities, ix + 211pp. 5% x 8.
S45 Paperbound $1.25
14
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NUMERICAL AND GRAPHICAL METHODS, TABLES
JACOB IAN ELLIPTIC FUNCTION TABLES, L. M. MilneThomson. Easytofollow, practical, not
only useful numerical tables, but complete elementary sketch of application of elliptic
functions. Covers description of principle properties; complete elliptic integrals; Fourier
series, expansions; periods, zeros, poles, residues, formulas for special values of argument;
cubic, quartic polynomials; pendulum problem; etc. Tables, graphs form body of book:
Graph, 5 figure table of elliptic function sn (u m); en (u m); dn (u m). 8 figure table of
complete elliptic integrals K, K', E, E', nome q. 7 figure table of Jacobian zetafunction
Z(u). 3 figures, xi + 123pp. 5% x 8. S194 Paperbound $1.35
TABLES OF FUNCTIONS WITH FORMULAE AND CURVES, E. Jahnke, F. Emde. Most comprehensive
1volume English text collection of tables, formulae, curves of transcendent functions. 4th
corrected edition, new 76page section giving tables, formulae for elementary functions not
in other English editions. Partial contents: sine, cosine, logarithmic integral; error integral;
elliptic integrals; theta functions; Legendre, Bessel, Riemann, Mathieu, hypergeometric
functions; etc. "Outoftheway functions for which we know no other source." Scientific
Computing Service, Ltd. 212 figures. 400pp. ; 5% x 8%. S133 Paperbound $2.00
MATHEMATICAL TABLES, H. B. Dwight. Covers in one volume almost every function of im
portance in applied mathematics, engineering, physical sciences. Three extremely fine
tables of the three trig functions, inverses, to 1000th of radian; natural, common logs;
squares, cubes, hyperbolic functions, inverses; (a 2 + b 2 ) exp. Via, complete elliptical in
tegrals of 1st, 2nd kind; sine, cosine integrals; exponential integrals; Ei(x) and Ei( — x);
binomial coefficients; factorials to 250; surface zonal harmonics, first derivatives; Bernoulli,
Euler numbers, their logs to base of 10; Gamma function; normal probability integral; over
60pp. Bessel functions; Riemann zeta function. Each table with formulae generally used,
sources of more extensive tables, interpolation data, etc. Over half have columns of
differences, to facilitate interpolation, viii + 231pp. 5% x 8. S445 Paperbound $1.75
PRACTICAL ANALYSIS, GRAPHICAL AND NUMERICAL METHODS, F. A. Willers. Immensely prac
tical handbook for engineers. How to interpolate, use various methods of numerical differ
entiation and integration, determine roots of a single algebraic equation, system of linear
equations, use empirical formulas, integrate differential equations, etc. Hundreds of short
cuts for arriving at numerical solutions. Special section on American calculating machines,
by T. W. Simpson. Translation by R. T. Beyer. 132 illustrations. 422pp. 5% x 8.
S273 Paperbound $2.00
NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS, H. Levy, E. A. Baggott. Comprehensive
collection of methods for solving ordinary differential equations of first and higher order
2 requirements: practical, easy to grasp; more rapid than school methods. Partial contents:
graphical integration of differential equations, graphical methods for detailed solution.
Numerical solution. Simultaneous equations and equations of 2nd and higher orders.
"Should be in the hands of all in research and applied mathematics, teaching," Nature.
21 figures, viii + 238pp. 5% x 8. S168 Paperbound $1.75
NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS, Bennet, Milne, Bateman. Unabridged
republication of original prepared for National Research Council. New methods of integration
by 3 leading mathematicians: "The Interpolational Polynomial," "Successive Approximation,"
A. A. Bennett, "Stepbystep Methods of Integration," W. W. Milne. "Methods for Partial
Differential Equations," H. Bateman. Methods for partial differential equations, solution
of differential equations to nonintegral values of a parameter will interest mathematicians,
physicists. 288 footnotes, mostly bibliographical. 235 item classified bibliography. 108pp.
5% X 8. S305 Paperbound $1.35
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THE BINARY STARS
BY ROBERT G. AITKEN
Originally published in 1918 and completely revised in 1935, this book is still a
"bible" in the field of doublestar astronomy. Its author, a director of the famous
Lick Observatory, is widely recognized as the father of the modern study of binary
star systems and, in this book, he sums up the results of centuries of research,
including 40 years of his own work, and surveys the methods of observation and of
orbital computation that have been developed.
The book begins with an extensive twopart historical sketch that covers various
steps in the discovery and interpretation of binary stars and assesses the work of
many of the major contributors. There follow chapters on observational methods for
visual binary stars; the orbit of a visual binary star; the radial velocity of a star
(written by Dr. J. K Moore of the University of California); the orbit of a spectro
scopic binary star; eclfpsing binary stars; the known orbits of the binary stars; some
binary stars of special interest: Alpha Centauri, Sirius, Krueger 60, 61 Cygni,
Delta Equulei, Capella, Algol, and others; and the origin of binary stars. Within each
of these topics, the author emphasizes method, telling how to plot orbits, where to
locate important data and how to use them in computations, how to measure and
reduce spectrograms, when to use eyepieces in observations, and so on. This practical
material is truly unique, and it should prove especially useful in classroom instruc
tion, to astronomy students, and to advanced hobbyists.
Each chapter ends with a bibliography and there is additional, uptodate bibli
ographical material in the notes added to this new edition by Professor J. T. Kent.
Professor Kent has also revised and corrected the text. Now, more than ever, this
book is an unequalled source of information on methods of study and as a thorough
summary of essential binary star material.
Revised (1935) edition, corrected and with additional notes new to 1963 edition by
Prof. J. T. Kent. New preface. 50 tables; 13 figures; 4 fullpage plates. Bibliographies.
Appendix: Table of Orbits of Visual and Spectroscopic Binary Stars. Indexes, x +
309pp. 5% x m. S1102 Paperbound $2.00
A DOVER EDITION DESIGNED FOR YEARS OF USE!
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